SRIT / UICM007 P & S / Two Dimensional Random Variables ...
Transcript of SRIT / UICM007 P & S / Two Dimensional Random Variables ...
SRIT / UICM007 – P & S / Two Dimensional Random Variables
SRIT / M & H / M. Vijaya Kumar 1
SRI RAMAKRISHNA INSTITUTE OF TECHNOLOGY
(AN AUTONOMOUS INSTITUTION)
COIMBATORE- 641010
UICM007 & PROBABILITY AND STATISTICS
Two Dimensional Random Variables
Let be the sample space of a random experiment . Let ( ) and ( ) be
two random variables each assigning real numbers to each outcome . Then ( ) is
called two dimensional random variable.
Types of random variable
Discrete random variable
Continuous random variable
Discrete random variable Continuous random variable
1 Joint Probability
[ ] ( ) ( ) ( )
2
Marginal Probability
, - ( ) ∑ ( )
[ ] ( ) ∑ ( )
( ) ( ) ∫ ( )
( ) ( ) ∫ ( )
3
Conditional Probability
[ ⁄ ] [ ]
( )
, ⁄ - [ ]
( )
( ) ( )
( )
( ) ( )
( )
4 If and are independent
[ ] ( ) ( ) ( ) ( ) ( )
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Problem: 1
( )
Find the marginal and conditional distributions.
Answer:
The marginal distributions of and are
Marginal probability
of is ( ) ( ) 1 2 3
1
( )
2
( )
Marginal probability of is
( ) ( ) ( )
( )
( )
∑ ( )
Conditional distribution of :
( ⁄ ) ( )
( )
1 2 3
1
2
Conditional distribution of :
( ⁄ ) ( )
( )
1 2 3
1
2
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Problem: 2
The joint probability distribution function is given by ( ) ( )
Find all the marginal and conditional probability distributions. Also
find the distribution function of .
Answer:
The marginal distributions are show below
( ) ( )
0 1 2
1 ( ) ( ) ( ) ( )
2 ( ) ( ) ( ) ( )
3 ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ∑ ( )
We know that the total probability
∑ ( )
The marginal distributions of and are
Marginal probability
of is ( ) 0 1 2
1
( )
2
( )
3
( )
Marginal Probability
of is ( ) ( )
( )
( )
∑ ( )
Conditional distribution of :
( ⁄ ) ( )
( )
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0 1 2
1
2
3
( )
( )
( )
( )
( )
( )
Conditional distribution of :
( ⁄ ) ( )
( )
0 1 2
1
( )
( )
2
( )
( )
3
( )
( )
Probability distribution function of
1 ( )
2 ( ) ( )
3 ( ) ( ) ( )
4 ( ) ( )
5 ( )
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Problem: 3
The bivariate random variable and has the probability density function
( ) ( )
Answer:
By definition of probability density function
∫ ∫ ( )
∫ ∫ ( )
∫ ∫ ( )
∫ *
+
∫ *( ( )
) (
)+
∫ *
+
∫ *
+
*
+
(
)
(
)
Problem: 4 The bivariate random variable and has the probability density function
( ) ( )( )
Answer:
By definition of probability density function
∫ ∫ ( )
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∫ ∫ ( )( )
∫ ( )
∫ ( )
*
+
*
+
, -
*
+ *(
) (
)+
, - [(
)
]
, - , -
Problem: 5
( ) { ( )
( ) ( ) ( ) ( ) .
/
Answer:
To find the constant C:
By definition of probability density function
∫ ∫ ( )
∫ ∫ ( )
∫ ∫ ( )
∫ *
+
∫ *(
) ( ( )
)+
∫ *
+
∫
*
+
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*
+
( ) ∫ ( )
∫ ( )
∫ ( )
[
]
*
+
*(
) ( ( )
)+
*
+
, -
( )
By definition of conditional probaility, we have
.
/ ( )
( )
( )
( )
.
/ ( )
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Problem: 6
The joint probability density function of two random variables and is given by
( ) {
i. Find the value ii. Find the mariginal density
functions of and .
Answer:
To find the constant :
By definition of probability density function
∫ ∫ ( )
∫ ∫
∫
∫
, -
(
)
(
)
(
) (
)
( ) ∫ ( )
∫
*
+
(
)
( ) ( )
( ) ∫ ( )
∫
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*
+
(
)
( )
Problem: 7
The joint probability density function of a two dimensional random variables is
( ) {
( ) (
)
( ) (
⁄ ) ( ) ( )
Answer:
( ) {
( ) ∫ ( )
∫ *
+
*
+
( )
( ) ∫ ( )
∫ *
+
*
+
( )
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(
) ∫ ( )
∫ [
]
(
)
(
)
(
)
(
)
(
⁄ )
. /
. /
(
) ∫ ∫ ( )
∫ ∫ *
+
∫ (
)
∫ (
(
)
(
))
(
)
(
) (
)
(
⁄ )
. /
. /
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( )
1
( ) ∫ ∫ (
)
∫ *
+
∫ *( )
( ) +
∫ [
( )
( ) ]
∫ [
( )
( ) ]
*
(
)
( )
( )+
[
(
)
( )]
[
(
)
( )]
( )
( )
𝑥 𝑥
𝑦
𝑦
𝑥 𝑦
𝑥 ( )
𝑦
( )
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Problem: 8
The joint probability density function of two random variables & is given by
( ) { (
)
(
) (
⁄ )
Answer:
( ) ∫ ( )
∫ (
)
*
+
, -
( )
( ) ∫ ( )
∫ (
)
(
) *
+
(
) [
]
( ) (
)
(
) ( )
( ) (
)
(
)
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(
)
(
)
Problem: 9
Determine whether the random variables and are independent, given their
( ) {
Also find the conditional densities of on and on .
Answer:
( ) ∫ ( )
∫ .
/
(
)
(
)
( )
( ) ∫ ( )
∫ .
/
(
)
(
)
( )
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( ) ( ) (
) (
)
( ) ( ) ( )
and are not independent.
(
) ( )
( )
(
)
.
/ ( )
( )
Problem: 10
If the joint PDF of the function and is given by
( ) {
( )
( ) ( ) ( ) ( ) ( ) ( )
Answer:
( ) ∫ ∫
( )
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∫ *
+
∫ *(
) (
)+
∫ [
]
∫ [
]
*
+
[
]
( )
( ) ( )
( )
( ) ∫ ∫
( )
∫ *
+
∫ *(
) (
)+
∫ [
]
∫ [
]
*
+
*
+
[
]
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( )
( ) ( )
( )
( )
( )
∫ ∫ ( )
∫ *
+
∫ * ( )
( )
( )+
∫ * ( )
( )
( )+
* ( )
( )
( )
( ) (
)+
* (
) (
(
))+
[
]
( )
( )
( )
( )
𝑥
𝑥
𝑦
𝑦
𝑥 𝑦
𝑥
𝑦
( )
𝑦
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Problem: 11
Let and have the joint p.d.f given by
( ) {
( )
( ) [(
) (
)] ( ) [(
) (
)]
( ) ( ) ( ) ( )
Answer:
( ) {
( ) ∫ ( )
∫
*
+
, -
( ) ( )
( ) ∫ ( )
∫
(
)
( )
( )
[(
) (
)] ∫ ∫
∫
∫
, -
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*
+
*
+
*(
)
+
*(
)
+
[
] [
]
[(
) (
)]
[(
)] 0.
/ .
/1
.
/
[(
) (
)] ∫ ∫
∫
∫
, -
*
+
*
+
*(
)
+
*(
)
(
)
+
[
] [
]
[
]
[(
) (
)]
(
) ∫ ( )
∫
*
+
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*(
)
(
)
+
(
)
(
) . /
. /
( ) ( ) ( ) ( )
and
( ) ∫ ( )
∫ ( )
∫ ( )
*
+
[
]
( )
( ) ∫ ( )
∫
∫
*
+
, -
( )
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Correlation
If the change in one variable affects the change in the other variable, then the
variable are said to be correlated.
Positive correlation:
If the two variable deviate in the same direction (i.e., increase or decrease) in one
variable in a corresponding (increase or decrease) in other variable is said to be positive
correlation.
Ex: Income and expenditure
Negative correlation:
If the two variable deviate in opposite direction (i.e., increase or decrease) in one
variable in a corresponding (decrease or increase) in other variable is said to be negative
correlation.
Ex: Price and demand of a product
Rank correlation
Sometimes there doesn’t exist a marked linear relationship between two random
variables but a monotonic relation (if one increases, the other also increases or instead,
decreases) is clearly noticed. Pearson’s Correlation Coefficient evaluation, in this case,
would give us the strength and direction of the linear association only between the
variables of interest. Herein comes the advantage of the Spearman Rank Correlation
methods, which will instead, give us the strength and direction of the monotonic relation
between the connected variables. This can be a good starting point for further evaluation.
The Spearman Rank-Order Correlation Coefficient
The Spearman’s Correlation Coefficient, represented by or by , is a
nonparametric measure of the strength and direction of the association that exists between
two ranked variables. It determines the degree to which a relationship is monotonic, i.e.,
whether there is a monotonic component of the association between two continuous or
ordered variables.
Monotonicity is “less restrictive” than that of a linear relationship. Although
monotonicity is not actually a requirement of Spearman’s correlation, it will not be
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meaningful to pursue Spearman’s correlation to determine the strength and direction of a
monotonic relationship if we already know the relationship between the two variables is
not monotonic.
Spearman Ranking of the Data
We must rank the data under consideration before proceeding with the Spearman’s
Rank Correlation evaluation. This is necessary because we need to compare whether on
increasing one variable, the other follows a monotonic relation (increases or decreases
regularly) with respect to it or not.
Thus, at every level, we need to compare the values of the two variables. The method of
ranking assigns such ‘levels’ to each value in the dataset so that we can easily compare it.
Assign number 1 to (the number of data points) corresponding to the variable
values in the order highest to lowest.
In the case of two or more values being identical, assign to them the arithmetic mean
of the ranks that they would have otherwise occupied.
For example, Selling Price values given: 28.2, 32.8, 19.4, 22.5, 20.0, 22.5 The
corresponding ranks are: 2, 1, 5, 3.5, 4, 3.5 The highest value 32.8 is given rank 1, 28.2 is
given rank 2,…. Two values are identical (22.5) and in this case, the arithmetic means of
ranks that they would have otherwise occupied (3+42) has to be taken.
Spearman’s Rank Correlation formula
∑
( )
where is the number of data points of the two variables and is the difference in the
ranks of the ith element of each random variable considered. The Spearman correlation
coefficient, , can take values from to .
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I. If the value of is indicates a perfect association of ranks
II. If the value of is indicates no association between ranks and
III. If the value of is indicates a perfect negative association of ranks.
IV. If the value of near to zero, the weaker the association between the ranks.
Merits of Rank Correlation Coefficient
1. Spearman’s rank correlation coefficient can be interpreted in the same way as the Karl
Pearson’s correlation coefficient;
2. It is easy to understand and easy to calculate;
3. If we want to see the association between qualitative characteristics, rank correlation
coefficient is the only formula;
4. Rank correlation coefficient is the non-parametric version of the Karl Pearson’s product
moment correlation coefficient; and
5. It does not require the assumption of the normality of the population from which the
sample observations are taken.
Demerits of Rank Correlation Coefficient
1. Product moment correlation coefficient can be calculated for bivariate frequency
distribution but rank correlation coefficient cannot be calculated; and
2. If , this formula is time consuming.
Problem: 12
The following table provides data about the percentage of students who have free
university meals and their CGPA scores. Calculate the Spearman’s Rank Correlation
between the two and interpret the result.
State University % of students having free meals % of students scoring above 8.5 CGPA
Pune 14.4 54
Chennai 7.2 64
Delhi 27.5 44
Kanpur 33.8 32
Ahmedabad 38.0 37
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Indore 15.9 68
Guwahati 4.9 62
Answer:
Let us first assign the random variables to the required data –
X – % of students having free meals
Y – % of students scoring above 8.5 CGPA
Before proceeding with the calculation, we’ll need to assign ranks to the data
corresponding to each state university. We construct the table for the rank as below
Rank in X Rank in Y
14.4
7.2
27.5
33.8
38.0
15.9
4.9
∑ ∑
Rank Correlation
∑
( )
(
( )) , -
(
)
The result shows that a strong negative coefficient of correlation. That is the highest
percentage of students consuming free meals tend to have the least successful results.
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Problem: 13
Compute the coefficient of rank correlation between sales and advertisement expressed in
thousands of dollars from the following data:
Sales 90 85 68 75 82 80 95 70
Advertisement 7 6 2 3 4 5 8 1
Answer:
Rank in X Rank in Y
90 2 7 2 0
85 3 6 3 0
68 8 2 7 1
75 6 3 6 0
82 4 4 5 1
80 5 5 4 1
95 1 8 1 0
70 7 1 8 1
∑ ∑
Rank Correlation
∑
( )
(
( )) , -
(
)
The result shows that a strong positive coefficient of correlation. Hence there is a very
good amount of agreement between sales and advertisement.
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Problem: 14
Find the rank correlation co efficient from the following data.
Rank in X 1 2 3 4 5 6 7
Rank in Y 4 3 1 2 6 5 7
Answer:
∑ ∑
Rank Correlation
∑
( )
(
( )) , -
(
)
Problem: 15
The ranks of some 16 students in mathematics and physics are as follows. Find the
rank correlation for the proficiency in mathematics and physics.
Rank in Math’s 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Rank in Physics 1 10 3 4 5 7 2 6 8 11 15 9 14 12 16 13
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Answer:
∑ ∑
Rank Correlation
∑
( )
(
( )) , -
(
)
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Problem: 16
Suppose we have ranks of 5 students in three subjects Computer, Physics and Statistics and
we want to test which two subjects have the same trend.
Rank in Computer 2 4 5 1 3
Rank in Physics 5 1 2 3 4
Rank in Statistics 2 3 5 4 1
Answer:
In this problem ranks are directly given.
Rank
in X
Rank
in Y
Rank
in Z
2 5 2
4 1 3
5 2 5
1 3 4
3 4 1
∑ ∑
∑
Rank Correlation
∑
( )
I. Rank correlation between computer and physics:
∑
( )
(
( )) , -
(
)
II. Rank correlation between physics and statistics:
∑
( )
(
( )) , -
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(
)
III. Rank correlation between computer and statistics:
∑
( )
(
( )) , -
(
)
Since and are negative which indicates that Computer and Physics also Physics and
Statistics have opposite trends. But indicates that Computer and Statistics have same
trend.
Repeated Ranks
If the value is repeated in any row in any series or series in times, then we
have add the correction factor in the rank correlation formula
( )
(∑
( ))
Problem: 17
Determine the rank correlation co efficient for the following data.
68 64 75 50 64 80 75 40 55 64
62 58 68 45 81 60 68 48 50 70
Answer:
Rank in Rank in
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∑ ∑
To find Correction Factor:
In series value 75 is repeated two times
( )
In series value 64 is repeated three times
( )
In series value 68 is repeated two times
( )
Rank Correlation:
(∑ ( )
)
(
( )) , -
(
)
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Problem: 18
The sample of 12 fathers and their eldest sons have the following data about their
heights in inches.
Fathers 65 63 67 64 68 62 70 66 68 67 69 71
Sons 68 66 68 65 69 66 68 65 71 67 68 70
Answer:
Rank in Rank in
∑ ∑
To find Correction Factor:
In series value 68 is repeated two times
( )
In series value 67 is repeated two times
( )
In series value 68 is repeated four times
( )
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In series value 66 is repeated two times
( )
In series value 65 is repeated two times
( )
Rank Correlation:
(∑
( ))
(
( )) , -
( )
Karl – pearson’s co efficient of Correlation
Karl Pearson’s Coefficient of Correlation is widely used mathematical method
wherein the numerical expression is used to calculate the degree and direction of the
relationship between linear related variables.
Pearson’s method, popularly known as a Pearsonian Coefficient of Correlation, is the
most extensively used quantitative methods in practice. The coefficient of correlation is
denoted by .
If the relationship between two variables and is to be ascertained, then the
following formula is used:
( ) ( ) ( )
( ) ( ) ( ) ( )
∑
∑
∑
( ) ( ) , ( )- ( )
( ) , ( )-
∑
∑
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Note:
If ( ) then ( ) ( ) ( )
If ( ) ( ) then and are uncorrelated.
Problem: 19
Find the covariance of and if the random variable ( ) having joint probability
density function ( ) {
Answer:
( ) ∫ ( )
∫ ( )
*
+
( ) (
)
( ) (
)
( ) ∫ ( )
∫ (
)
∫ .
/
*
+
[
]
( )
( )
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( ) ∫ ∫ ( )
∫ ∫ ( )
∫ ∫ ( )
∫ *
+
∫ *
+
*
+
( )
( ) ( ) ( ) ( )
Problem: 20
The joint probability density function of a random variable ( ) is
( ) and Find the covariance of and .
Answer:
( ) ∫ ( )
∫
*
+
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*
+ , -
( )
( ) ∫ ( )
∫
(
)
(
)
( )
( ) ∫ ( )
∫
, -
( )
( )
( ) ∫ ( )
∫
(
( ) )
( *
+)
( )
( ) ∫ ∫ ( )
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∫ ∫
∫ (
( ) )
, -
∫ ( *
+)
∫
(
)
(
)
( )
( ) ( ) ( ) ( )
Since Covariance of and is 0. Therefore and are independent.
Problem: 21
Two random variables and having the joint probability density function given by
( ) { ( )
i. Find the value ii. Find the marginal
density function of and iii. Find the correlation co efficient of and .
Answer:
I. To find the value :
By definition of total probability density function
∫ ∫ ( )
∫ ∫ ( )
∫ (
)
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∫ (
)
(
)
(
)
( ) ∫ ( )
∫ ( )
*
+
( )
(
)
( ) ∫ ( )
∫ ( )
*
+
( )
.
/
( ) ∫ ( )
∫
(
)
∫ (
)
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(
)
(
)
( )
( ) ∫ ( )
∫
(
)
∫ (
)
(
)
(
)
( )
( ) ∫ ( )
∫
.
/
∫ (
)
(
)
(
)
( )
( ) ∫ ( )
∫
.
/
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∫ (
)
(
)
(
)
( )
( ) ∫ ∫ ( )
∫ ∫ ( )
∫ ∫ ( )
∫ (
)
∫ (
)
(
)
(
)
( )
√ ( ) * ( )+
√
(
)
√ ( ) * ( )+
√
(
)
( ) ( ) ( ) ( )
(
)
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( ) ( )
( )
Problem: 22
The joint probability density function of a two dimensional random variables are
( ) ( ) ( )
Answer:
( ) ( )
1
( ) ∫ ∫ ( )
∫ ∫ ( )
∫ ∫ ( )
∫ *
+
∫ *( )
( ) +
∫ *( )
( ) +
∫ *( )
( )+
( )
𝑥
𝑥
𝑦
𝑦
𝑥 𝑦
𝑥 ( )
𝑦
( )
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*( )
( )
(
)+
[
(
) (
)]
(
)
( )
( )
( ) ∫ ∫ ( )
∫ ∫ ( )
∫ ∫ ( )
∫ *
+
∫ *( )
( ) +
∫ *( )
( ) +
∫ *( )
( )+
*( )
( )
(
)+
[
(
) (
)]
(
)
( )
( )
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SRIT / M & H / M. Vijaya Kumar 41
( ) ∫ ∫ ( )
∫ ∫ ( )
∫ ∫ ( )
∫ *
+
∫ *( )
( ) +
∫ *( )
( ) +
∫ [
( )
( )]
*
(
)
(
)+
[
(
)
(
)]
(
)
( )
√ ( ) * ( )+
√
(
)
( ) ( ) ( ) ( )
SRIT / UICM007 – P & S / Two Dimensional Random Variables
SRIT / M & H / M. Vijaya Kumar 42
( ) ( )
( )
Problem: 23
Let and be random variables having joint probability density function
( ) {
( )
Answer:
( ) ∫ ( )
∫ ( )
*
+
[
]
( )
( )
( ) ∫ ( )
∫ (
)
∫ ( )
SRIT / UICM007 – P & S / Two Dimensional Random Variables
SRIT / M & H / M. Vijaya Kumar 43
*
+
[
]
( )
( )
( ) ∫ ( )
∫ (
)
∫ ( )
*
+
[
]
( )
( )
( ) ∫ ∫ ( )
∫ ∫
( )
∫ ∫ ( )
∫ *
+
SRIT / UICM007 – P & S / Two Dimensional Random Variables
SRIT / M & H / M. Vijaya Kumar 44
∫ [
]
*
+
[
]
( )
√ ( ) * ( )+
√
(
)
( ) ( ) ( ) ( )
( ) ( )
( )
Problem: 24
Let and be random variables having joint probability density function
( ) { ( )
Answer:
To find the constant C:
We know that the total probability
∫ ∫ ( )
∫ ∫ ( )
SRIT / UICM007 – P & S / Two Dimensional Random Variables
SRIT / M & H / M. Vijaya Kumar 45
∫ *
+
∫ (
)
*
+
*
+
( ) ∫ ( )
∫ ( )
*
+
*
+
, -
, -
( )
( )
( ) ∫ ( )
∫ (
)
∫ ( )
SRIT / UICM007 – P & S / Two Dimensional Random Variables
SRIT / M & H / M. Vijaya Kumar 46
*
+
[
]
( )
( )
( ) ∫ ( )
∫ (
)
∫ ( )
*
+
*
+
[
]
( )
( )
( ) ∫ ∫ ( )
∫ ∫ ( )
∫ ∫ ( )
∫ *
+
∫ *
+
SRIT / UICM007 – P & S / Two Dimensional Random Variables
SRIT / M & H / M. Vijaya Kumar 47
*
+
*
+
[
]
( )
√ ( ) * ( )+
√ (
)
( ) ( ) ( ) ( )
( ) ( )
( )
Problem: 25
Let and be random variables having joint probability density function
( ) { ,( ) ( )- ( )
Show that and are
uncorrelated but not independent.
Answer:
( ) { , -
To find the constant :
SRIT / UICM007 – P & S / Two Dimensional Random Variables
SRIT / M & H / M. Vijaya Kumar 48
We know that the total probability
∫ ∫ ( )
∫ ∫ , -
∫ *
+
∫ [
]
*
+
[
]
(
)
( ) ∫ ( )
∫ , -
*
+
[
]
( ) [
]
( ) [
]
( ) ∫ ( )
∫ [
]
∫ (
)
SRIT / UICM007 – P & S / Two Dimensional Random Variables
SRIT / M & H / M. Vijaya Kumar 49
(
)
(
)
( )
( )
( ) ∫ ∫ ( )
∫ ∫ ( )
∫ ∫ ( )
∫ (
)
∫ (
)
(
)
(
)
(
)
(
)
( )
( ) ( ) ( ) ( )
and are uncorrelated.
SRIT / UICM007 – P & S / Two Dimensional Random Variables
SRIT / M & H / M. Vijaya Kumar 50
( ) ( ) [
] [
] ( )
and are not independent.
Problem: 26
The joint pdf of and is ( ) { ( )
Find covariance of
and . Are and are independent?
Answer:
( ) {
( ) ∫ ( )
∫
[
]
(
)
(
) , -
( )
( ) ∫ ( )
∫
[
]
(
)
(
) , -
( )
SRIT / UICM007 – P & S / Two Dimensional Random Variables
SRIT / M & H / M. Vijaya Kumar 51
( ) ∫ ( )
∫
(
( ) )
, -
,( ) ( )- , -
( )
( )
( ) ∫ ∫ ( )
∫ ∫
∫
∫
, -
(
( ) )
(
( ) )
,( ) ( )- ,( ) ( )-
( )
( ) ( ) ( ) ( )
and are uncorrelated.
( ) ( ) ( )
and are independent.
Problem: 27
Find the correlation co efficient for the following data
X 10 14 18 22 26 30
Y 18 12 24 6 30 36
Answer:
SRIT / UICM007 – P & S / Two Dimensional Random Variables
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10 18 180 100 324
14 12 168 196 144
18 24 432 324 576
22 6 132 484 36
26 30 780 676 900
30 36 1080 900 1296
∑ ∑ ∑ ∑ ∑
∑
∑
√∑
( )
√
( )
√∑
( )
√
( )
( ) ∑
( ) ( )
Problem: 28
The table below shows the number of absences, , in a Calculus course and the final exam
grade, , for 7 students. Find the correlation coefficient.
X 1 0 2 6 4 3 3 Y 95 90 90 55 70 80 85
Answer:
SRIT / UICM007 – P & S / Two Dimensional Random Variables
SRIT / M & H / M. Vijaya Kumar 53
1 95 1 9025 95
0 90 0 8100 0
2 90 4 8100 180
6 55 36 3025 330
4 70 16 4900 280
3 80 9 6400 240
3 85 9 7225 255
∑ ∑ ∑ ∑ ∑
∑
∑
√∑
( ) √
√∑
( ) √
( ) ∑
( ) ( )
Problem: 29
The marks obtained by 10 students in Mathematics and Statistics are given below.
Find the correlation coefficient between the two subjects.
Marks in math’s 75 30 60 80 53 35 15 40 38 48
Marks in Stats 85 45 54 91 58 63 35 43 45 44
Answer:
SRIT / UICM007 – P & S / Two Dimensional Random Variables
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75 85 6375 5625 7225
30 45 1350 900 2025
60 54 3240 3600 2916
80 91 7280 6400 8281
53 58 3074 2809 3364
35 63 2205 1225 3969
15 35 525 225 1225
40 43 1720 1600 1849
38 45 1710 1444 2025
48 44 2112 2304 1936
∑ ∑ ∑ ∑ ∑
∑
∑
√∑
( ) √
√∑
( ) √
( ) ∑
( ) ( )
Problem: 30
Compute the coefficient of correlation between X and Y using the following data:
SRIT / UICM007 – P & S / Two Dimensional Random Variables
SRIT / M & H / M. Vijaya Kumar 55
X 1 3 5 7 8 10
Y 8 12 15 17 18 20
Answer:
1 8 8 1 64
3 12 36 9 144
5 15 75 25 225
7 17 119 49 289
8 18 144 64 324
10 20 200 100 400
∑ ∑ ∑ ∑ ∑
∑
∑
√∑
( )
√
√∑
( )
√
( ) ∑
( ) ( )
SRIT / UICM007 – P & S / Two Dimensional Random Variables
SRIT / M & H / M. Vijaya Kumar 56
Problem: 31
Calculate the coefficient of correlation for the following data:
9 8 7 6 5 4 3 2 1
15 16 14 13 11 12 10 8 9
Answer:
9 15 135 81 225
8 16 128 64 256
7 14 98 49 196
6 13 78 36 169
5 11 55 25 121
4 12 48 16 144
3 10 30 9 100
2 8 16 4 64
1 9 9 1 81
∑ ∑ ∑ ∑ ∑
∑
∑
√∑
( )
√
√∑
( )
√
( ) ∑
SRIT / UICM007 – P & S / Two Dimensional Random Variables
SRIT / M & H / M. Vijaya Kumar 57
( ) ( )
Regression
Regression analysis is most often used for prediction. The goal in regression analysis
is to create a mathematical model that can be used to predict the values of a dependent
variable based upon the values of an independent variable. In other words, we use the
model to predict the value of when we know the value of . (The dependent variable is
the one to be predicted). Correlation analysis is often used with regression analysis
because correlation analysis is used to measure the strength of association between the
two variables and .
In regression analysis involving one independent variable and one dependent
variable the values are frequently plotted in two dimensions as a scatter plot. The scatter
plot allows us to visually inspect the data prior to running a regression analysis. Often this
step allows us to see if the relationship between the two variables is increasing or
decreasing and gives only a rough idea of the relationship. The simplest relationship
between two variables is a straight-line or linear relationship. Of course the data may well
be curvilinear and in that case we would have to use a different model to describe the
relationship (we will deal only with linear relationship’s for now). Simple linear regression
analysis finds the straight line that best fits the data
Definition:
Regression is mathematical measure of the average relationship between two or
more variables in terms of original limits of the data.
The equation of line of regression of on is
( )
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The equation of line of regression of on is
( )
Regression co efficient
Correlation co efficient √
Problem: 32
Obtain the equations of the lines of regression from the following data:
1 2 3 4 5 6 7
9 8 10 12 11 13 14
Answer:
1 9 9 1 81
2 8 16 4 64
3 10 30 9 100
4 12 48 16 144
5 11 55 25 121
6 13 78 36 169
7 14 98 49 196
Total 334 140 875
∑
∑
√∑
( ) √
√∑
( ) √
SRIT / UICM007 – P & S / Two Dimensional Random Variables
SRIT / M & H / M. Vijaya Kumar 59
( ) ( ) ( ) ( )
∑
Correlation coefficient
( )
The line of regression of on is
( )
( )
The line of regression of on is
( )
( )
Problem: 33
From the following data find
The two regression lines
The co efficient of correlation between the marks in economics and statistics
The most likely marks in statistics when marks in economics are 30.
Marks in Economics: 25 28 35 32 31 36 29 38 34 32
Marks in Statistics: 43 46 49 41 36 32 31 30 33 39
Answer:
25 43 1075 625 1849
28 46 1288 784 2116
35 49 1715 1225 2401
32 41 1312 1024 1681
31 36 1116 961 1296
36 32 1152 1296 1024
29 31 899 841 961
38 30 1140 1444 900
SRIT / UICM007 – P & S / Two Dimensional Random Variables
SRIT / M & H / M. Vijaya Kumar 60
34 33 1122 1156 1089
32 39 1248 1024 1521
Total 12067 10380 14838
∑
∑
√∑
( )
√
√∑
( )
√
( ) ( ) ( ) ( )
∑
Correlation coefficient
( )
The line of regression of on is
( )
( )
( )
The line of regression of on is
( )
( )
( )
The most likely marks in statistics when marks in economics are 30.
( )
Problem: 34
SRIT / UICM007 – P & S / Two Dimensional Random Variables
SRIT / M & H / M. Vijaya Kumar 61
A tyre manufacturing company is interested in removing pollutants from the exhaust at the
factory, and cost is a concern. The company has collected data from other companies
concerning the amount of money spent on environmental measures and the resulting
amount of dangerous pollutants released (as a percentage of total emissions)
Money
spent
(Rupees in
lakhs
8.4 10.2 16.5 21.7 9.4 8.3 11.5 18.4 16.7 19.3 28.4 4.7 12.3
Percentage
of
dangerous
pollutants
35.9 31.8 24.7 25.2 36.8 35.8 33.4 25.4 31.4 27.4 15.8 31.5 28.9
a) Compute the regression equation.
b) Predict the percentage of dangerous pollutants released when Rs. 20,000 is spent on
control measures.
Answer:
S. No
1 8.4 35.9 70.56 1288.8 301.56
2 10.2 31.8 104.04 1011.2 324.36
3 16.5 24.7 272.25 610.09 407.55
4 21.7 25.2 470.89 635.04 546.84
5 9.4 36.8 88.36 1354.2 345.92
6 8.3 35.8 68.89 1281.6 297.14
7 11.5 33.4 132.25 1115.6 384.1
8 18.4 25.4 338.56 645.16 467.36
9 16.7 31.4 278.89 985.96 524.38
10 19.3 27.4 372.49 750.76 528.82
11 28.4 15.8 806.56 249.64 448.72
12 4.7 31.5 22.09 992.25 148.05
13 12.3 28.9 151.29 835.21 355.47
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SRIT / M & H / M. Vijaya Kumar 62
Total ∑ 185.8 ∑ 384 ∑ 3177 ∑ 11756 ∑ 5080
a) Regression equation
∑
∑
√∑
( )
√
√∑
( )
√
( ) ( ) ( ) ( )
∑
Correlation coefficient
( )
The line of regression of on is
( )
( )
( )
The line of regression of on is
( )
( )
( )
b) When Rs. 20,000 ( ) is spent on control then the percentage of
dangerous pollutants released is
( )
Problem: 35
The quantity of a raw material purchased by a company at the specified prices during the
12 months of 1992 is given
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SRIT / M & H / M. Vijaya Kumar 63
MONTH PRICE/KG QUANTITY (KG)
Jan 96 250
Feb 110 200
Mar 100 250
Aprl 90 280
May 86 300
June 92 300
July 112 220
Aug 112 220
Sep 108 200
Oct 116 210
Nov 86 300
Dec 92 250
Find the regression equation based on the above data
Can you estimate the appropriate quantity likely to be purchased if the price shoot
upon Rs 124/kg?
Hence or otherwise obtain the coefficient of correlation between the price prevailing
and the quantity demanded
Answer:
S. No
1 96 250 9216 62500 24000
2 110 200 12100 40000 22000
3 100 250 10000 62500 25000
4 90 280 8100 78400 25200
5 86 300 7396 90000 25800
6 92 300 8464 90000 27600
7 112 220 12544 48400 24640
8 112 220 12544 48400 24640
9 108 200 11664 40000 21600
10 116 210 13456 44100 24360
11 86 300 7396 90000 25800
SRIT / UICM007 – P & S / Two Dimensional Random Variables
SRIT / M & H / M. Vijaya Kumar 64
12 92 250 8464 62500 23000
Total ∑ 1200 ∑ 2980 ∑ 121344 ∑ 756800 ∑ 293640
a) Regression equation
∑
∑
√∑
( )
√
√∑
( )
√
( ) ( ) ( ) ( )
∑
Correlation coefficient
( )
The line of regression of on is
( )
( )
( )
The line of regression of on is
( )
( )
( )
b) Estimation of the appropriate quantity likely to be purchased if the price shoot upon
Rs 124/kg,
Given price short , then
( )
c) Correlation coefficient
SRIT / UICM007 – P & S / Two Dimensional Random Variables
SRIT / M & H / M. Vijaya Kumar 65
Problem: 36
The two lines of regression are . The
variance of is 9. Find (i) The mean values of and (ii) correlation coefficient between
and .
Answer:
Since both the lines of regression passes through the mean values and .
The point ( ) must satisfy the two lines.
( )
( )
Solving ( ) and ( ), we get
The mean values of and are
Consider the line is a regression line on .
Consider the line is a regression line on .
Correlation co efficient √
√
√
SRIT / UICM007 – P & S / Two Dimensional Random Variables
SRIT / M & H / M. Vijaya Kumar 66
Correlation co efficient
Problem: 37
The two lines of regression are . Find ( ) and
correlation coefficient between and .
Answer:
Since both the lines of regression passes through the mean values and .
The point ( ) must satisfy the two lines.
( )
( )
Solving ( ) and ( ), we get
Consider the line is a regression line on .
Consider the line is a regression line on .
Correlation co efficient √
√
√
SRIT / UICM007 – P & S / Two Dimensional Random Variables
SRIT / M & H / M. Vijaya Kumar 67
Correlation co efficient
Problem: 38
The regression equation of and is . If the mean value of
and the correlation coefficient.
Answer:
Given the regression equation of and is
.
Since the line of regression passes through ( ), then
Also given mean value of is
( )
Hence mean value of is 48.
which is the line of regression of on .
[
]
SRIT / UICM007 – P & S / Two Dimensional Random Variables
SRIT / M & H / M. Vijaya Kumar 68
( )
( )
. /
Problem: 39
If and are uncorrelated random variables with variances and . Find the
correlation coefficient between and .
Answer:
Given that ( ) ( )
Let us take and
Given that both and are uncorrelated.
Now ( ) ( )
( ) ( ) , ( ) ( ) ( )-
( )
and ( ) ( )
( ) ( ) , ( ) ( ) ( )-
( )
( ) ,( )( )-
( )
SRIT / UICM007 – P & S / Two Dimensional Random Variables
SRIT / M & H / M. Vijaya Kumar 69
( ) ( )
( ) ,( )- ( ) ( )
( ) ,( )- ( ) ( )
Now ( ) ( ) ( ) ( )
( ) ( ) * ( ) ( )+ * ( ) ( )+
( ) ( ) ,* ( )+ * ( )+ -
, ( ) * ( )+ - , ( ) * ( )+ -
( ) ( )
( ) ( )
Problem: 40
If the independent random variables and have the variances 3 and
respectively, find the correlation coefficient between and .
Answer:
Given that ( ) ( )
Let us take and
Given that both and are uncorrelated.
Now ( ) ( )
( ) ( ) , ( ) ( ) ( )-
( )
√
and ( ) ( )
( ) ( ) , ( ) ( ) ( )-
( )
√
( ) ,( )( )-
SRIT / UICM007 – P & S / Two Dimensional Random Variables
SRIT / M & H / M. Vijaya Kumar 70
( )
( ) ( )
( ) ,( )- ( ) ( )
( ) ,( )- ( ) ( )
Now ( ) ( ) ( ) ( )
( ) ( ) * ( ) ( )+ * ( ) ( )+
( ) ( ) ,* ( )+ * ( )+ -
, ( ) * ( )+ - , ( ) * ( )+ -
( ) ( )
( ) ( )
√ √
Two Marks
1. Define the joint pmf of a two-dimensional discrete random variable.
Answer:
The function ( ) ( ) is called the joint pmf of a two-
dimensional discrete random variable and and and is denoted by .
2. The joint pmf of two random variables X and Y is given by
( ) {
Determine the value of the constant k .
Answer:
The marginal distributions are show below
1 2 3
1 k 2k 3k
2 2k 4k 6k
3 3k 6k 9k
We know that the total probability
∑ ( )
SRIT / UICM007 – P & S / Two Dimensional Random Variables
SRIT / M & H / M. Vijaya Kumar 71
3. Let X and Y be two discrete random variables with joint probability mass
( ) {
( )
probability mass functions of X and Y .
Answer:
The marginal distributions are show below
( ) ( )
1 2
1 3/18 5/18 ( )
2 4/18 6/18 ( )
( ) ( ) ( ) ( )
4. The joint probability mass function of a two dimensional random variable ( ) is
given by ( ) ( ) , where is constant. Find the value .
Answer:
( ) ( )
1 2
1 3k 5k ( )
2 4k 6k ( )
∑ ( )
We know that the total probability
∑ ( )
5. Find the value of , if ( ) ( )( ) and ( ) ,
otherwise, is to be the joint density function.
Answer:
By definition of total probability density function
SRIT / UICM007 – P & S / Two Dimensional Random Variables
SRIT / M & H / M. Vijaya Kumar 72
∫ ∫ ( )
∫ ∫ ( )( )
∫ ( )
∫ ( )
*
+
*
+
, -
[(
) ] [(
) ]
[
] [
]
6. Let and be continuous random variables with joint probability density function
( ) ( )
( ) ⁄ .
/
Answer:
.
/ ( )
( )
( ) ( ) ∫ ( )
∫ ( )
*
+
*(
) ( ( )
)+
*
+
, -
( )
SRIT / UICM007 – P & S / Two Dimensional Random Variables
SRIT / M & H / M. Vijaya Kumar 73
.
/ ( )
( )
( )
( )
.
/
( ) ( )
. Find ( ).
Answer:
( )
1
0 1
( ) ∫ ∫ (
)
∫ *
+
∫ *
+
∫ [
]
*
+
( )
( )
𝑥 𝑥
𝑦
𝑦
𝑥 𝑦
𝑥 ( )
𝑦
( )
SRIT / UICM007 – P & S / Two Dimensional Random Variables
SRIT / M & H / M. Vijaya Kumar 74
8. Find the marginal density functions of and if
( ) {
( )
Answer:
( ) ∫ ( )
∫ ( )
(
)
(
)
( ) ∫ ( )
∫ ( )
(
)
(
)
9. Find the acute angle between the two lines of regression, assuming the two
lines of regression.
Answer:
(
)
10. If the joint pdf of ( ) is ( ) { ( )
Check whether
and are independent? Answer:
( ) ∫ ( )
SRIT / UICM007 – P & S / Two Dimensional Random Variables
SRIT / M & H / M. Vijaya Kumar 75
∫
[
]
(
)
(
)
( )
( )
( ) ( )
( )
( )
and are independent.
11. The joint pdf of the random variable ( ) is given by ( ) ( )
. Find the value of .
Answer:
We know that the total probability
∫ ∫ ( )
∫ ∫ ( )
∫
∫
, -
Take ; When and
∫
∫
(
)
(
)
(
) (
)
(
) (
)
12. The two regression equations of two random variables and are
SRIT / UICM007 – P & S / Two Dimensional Random Variables
SRIT / M & H / M. Vijaya Kumar 76
and . Find the mean values of and .
Answer:
Since both the lines of regression passes through the mean values and .
The point ( ) must satisfy the two lines.
( )
( )
Solving ( ) and ( ), we get
Questions Problem: 1
( )
Find the marginal and conditional distributions.
Problem: 2
The joint probability distribution function is given by ( ) ( )
Find all the marginal and conditional probability distributions. Also
find the distribution function of .
Problem: 3
The bivariate random variable and has the probability density function
( ) ( )
Problem: 4 The bivariate random variable and has the probability density function
( ) ( )( )
Problem: 5
( ) { ( )
( ) ( ) ( ) ( ) .
/
Problem: 6
The joint probability density function of two random variables and is given by
SRIT / UICM007 – P & S / Two Dimensional Random Variables
SRIT / M & H / M. Vijaya Kumar 77
( ) {
i. Find the value ii. Find the mariginal density
functions of and .
Problem: 7
The joint probability density function of a two dimensional random variables is
( ) {
( ) (
)
( ) (
⁄ ) ( ) ( )
Problem: 8
The joint probability density function of two random variables & is given by
( ) { (
)
(
) (
⁄ )
Problem: 9
Determine whether the random variables and are independent, given their
( ) ,
Also find the conditional densities of on and on .
Problem: 10
If the joint PDF of the function and is given by
( ) {
( )
( ) ( ) ( ) ( ) ( ) ( )
Problem: 11
Let and have the joint p.d.f given by
( ) {
( )
( ) [(
) (
)] ( ) [(
) (
)]
( ) ( ) ( ) ( )
SRIT / UICM007 – P & S / Two Dimensional Random Variables
SRIT / M & H / M. Vijaya Kumar 78
Problem: 12
The following table provides data about the percentage of students who have free
university meals and their CGPA scores. Calculate the Spearman’s Rank Correlation
between the two and interpret the result.
State University % of students having free meals % of students scoring above 8.5 CGPA
Pune 14.4 54 Chennai 7.2 64
Delhi 27.5 44 Kanpur 33.8 32
Ahmedabad 38.0 37 Indore 15.9 68
Guwahati 4.9 62 Problem: 13
Compute the coefficient of rank correlation between sales and advertisement expressed in
thousands of dollars from the following data:
Sales 90 85 68 75 82 80 95 70
Advertisement 7 6 2 3 4 5 8 1
Problem: 14
Find the rank correlation co efficient from the following data.
Rank in X 1 2 3 4 5 6 7
Rank in Y 4 3 1 2 6 5 7
Problem: 15
The ranks of some 16 students in mathematics and physics are as follows. Find the
rank correlation for the proficiency in mathematics and physics.
Rank in Math’s 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Rank in Physics 1 10 3 4 5 7 2 6 8 11 15 9 14 12 16 13
Problem: 16
Suppose we have ranks of 5 students in three subjects Computer, Physics and Statistics and
we want to test which two subjects have the same trend.
Rank in Computer 2 4 5 1 3
Rank in Physics 5 1 2 3 4
Rank in Statistics 2 3 5 4 1
Problem: 17
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SRIT / M & H / M. Vijaya Kumar 79
Determine the rank correlation co efficient for the following data.
68 64 75 50 64 80 75 40 55 64
62 58 68 45 81 60 68 48 50 70
Problem: 18
The sample of 12 fathers and their eldest sons have the following data about their
heights in inches.
Fathers 65 63 67 64 68 62 70 66 68 67 69 71
Sons 68 66 68 65 69 66 68 65 71 67 68 70
Problem: 19
Find the covariance of and if the random variable ( ) having joint probability
density function ( ) {
Problem: 20
The joint probability density function of a random variable ( ) is
( ) and Find the covariance of and .
Problem: 21
Two random variables and having the joint probability density function given by
( ) { ( )
i. Find the value ii. Find the marginal
density function of and iii. Find the correlation co efficient of and .
Problem: 22
The joint probability density function of a two dimensional random variables are
( ) ( ) ( )
Problem: 23
Let and be random variables having joint probability density function
( ) {
( )
Problem: 24
Let and be random variables having joint probability density function
( ) { ( )
Problem: 25
SRIT / UICM007 – P & S / Two Dimensional Random Variables
SRIT / M & H / M. Vijaya Kumar 80
Let and be random variables having joint probability density function
( ) { ,( ) ( )- ( )
Show that and are
uncorrelated but not independent. Problem: 26
The joint pdf of and is ( ) { ( )
Find covariance of
and . Are and are independent?
Problem: 27
Find the correlation co efficient for the following data
X 10 14 18 22 26 30
Y 18 12 24 6 30 36
Problem: 28
The table below shows the number of absences, , in a Calculus course and the final exam
grade, , for 7 students. Find the correlation coefficient.
X 1 0 2 6 4 3 3
Y 95 90 90 55 70 80 85
Problem: 29
The marks obtained by 10 students in Mathematics and Statistics are given below.
Find the correlation coefficient between the two subjects.
Marks in math’s 75 30 60 80 53 35 15 40 38 48
Marks in Stats 85 45 54 91 58 63 35 43 45 44
Problem: 30
Compute the coefficient of correlation between X and Y using the following data:
X 1 3 5 7 8 10
Y 8 12 15 17 18 20
Problem: 31
Calculate the coefficient of correlation for the following data:
9 8 7 6 5 4 3 2 1
15 16 14 13 11 12 10 8 9
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Problem: 32
Obtain the equations of the lines of regression from the following data:
1 2 3 4 5 6 7
9 8 10 12 11 13 14
Problem: 33
From the following data find
The two regression lines
The co efficient of correlation between the marks in economics and statistics
The most likely marks in statistics when marks in economics are 30.
Marks in Economics: 25 28 35 32 31 36 29 38 34 32
Marks in Statistics: 43 46 49 41 36 32 31 30 33 39
Problem: 34
A tyre manufacturing company is interested in removing pollutants from the exhaust at the
factory, and cost is a concern. The company has collected data from other companies
concerning the amount of money spent on environmental measures and the resulting
amount of dangerous pollutants released (as a percentage of total emissions)
Money spent
(Rupees in lakhs
8.4 10.2 16.5 21.7 9.4 8.3 11.5 18.4 16.7 19.3 28.4 4.7 12.3
Percentage of
dangerous pollutants
35.9 31.8 24.7 25.2 36.8 35.8 33.4 25.4 31.4 27.4 15.8 31.5 28.9
a) Compute the regression equation.
b) Predict the percentage of dangerous pollutants released when Rs. 20,000 is spent on
control measures.
Problem: 35
The quantity of a raw material purchased by a company at the specified prices during the
12 months of 1992 is given
MONTH PRICE/KG QUANTITY (KG) Jan 96 250 Feb 110 200
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Mar 100 250 Aprl 90 280 May 86 300 June 92 300 July 112 220 Aug 112 220 Sep 108 200 Oct 116 210 Nov 86 300 Dec 92 250
Find the regression equation based on the above data
Can you estimate the appropriate quantity likely to be purchased if the price shoot
upon Rs 124/kg?
Hence or otherwise obtain the coefficient of correlation between the price prevailing
and the quantity demanded
Problem: 36
The two lines of regression are . The
variance of is 9. Find (i) The mean values of and (ii) correlation coefficient between
and .
Problem: 37
The two lines of regression are . Find ( ) and
correlation coefficient between and .
Problem: 38
The regression equation of and is . If the mean value of
and the correlation coefficient.
Problem: 39
If and are uncorrelated random variables with variances and . Find the
correlation coefficient between and .
Problem: 40
If the independent random variables and have the variances 3 and
respectively, find the correlation coefficient between and .
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“The best brains of the nation may be found on
the last benches of the classroom.”
― Dr. A. P. J. Abdul Kalam