Probability distributions
• Discrete
– Binomial distribution
– Poisson distribution
• Continuous
– Normal distribution
• Discrete random variable
– Variable is the characteristic of interest that
assumes different values for different elements of
the sample/population.
– If the value of the variable depends on the outcome
of an experiment it is called a random variable.
– Discrete random variable takes on a countable
number of values.
• Discrete distribution function - example
– Toss a coin twice.
– S = {HH; HT; TH; TT}
– Each outcome in S has a probability of ¼.
– Random variable X – number of heads
– Collection of probabilities – probability distribution
– associates a probability with each value of
random variable.
x 0 1 2
P(X = x) = P(x) 2
4
1
4
1
4
• Discrete distribution function
– 0 ≤ P(x) ≤ 1, for each x
– ∑P(x) = 1
3
4
4
4
1
4
x 0 1 2
P(X = x)
P(X ≤ x)
2
4
1
4
1
4+ + = 1
x 0 1 2 3 4
P(x) 0.15 0.3 0.25 0.2 0.1
Use the probability distribution given above to calculate:-
1. The probability that exactly 3 memory chips are
returned
2. The probability that more than two memory chips are
returned
3. That at least two memory chips are returned
4. From 1-3 memory chips are returned
5. Less than 2 memory chips are returned
6. At the most 2 memory chips are returned
7. Between one and four memory chips are returned
Let X denote the number of defective memory chips that are
returned to the production plant in a production batch of 300.The
number of returns received varies from 0 – 4.
Answers
• Example 5.3, p154 Elementary
Statistics
MEAN
• Represents average value that we expect to
obtain if the experiment is performed a large
number of times
( ) ( )E X xP x STANDARD DEVIATION
2 2( )x P x
• SD gives a measure of how dispersed
around the mean the variable is
• Discrete distribution function
– Mean = – expected value
– St dev =
0 1 2
P(X = x) 2
4
1
4
1
4
2 2( )x P x
( ) ( )E X xP x
1 2 1( ) 0 1 2 1
4 4 4xP x
1 2 12 2 2 2 2 2
4 4 4( ) 0 1 2 1 0.71x P x
• Discrete distribution function - example
– A survey was done to determine the number of
vehicles in a household.
– A sample of 560 households was taken and the
number of cars was captured.
– Random variable X – number of cars.
– The results are:
x – Number of cars 0 1 2 3 4
Number of
households
28 168 252 79 33
• Discrete distribution function - example
x – Number of cars 0 1 2 3 4
Number of
households 28 168 252 79 33
P(X = x) 0.05 0.30 0.45 0.14 0.06
28( 0) 0.05
560P X
168( 1) 0.30
560P X
252( 2) 0.45
560P X
560
• Discrete distribution function - example
x – Number of cars 0 1 2 3 4
Number of
households 28 168 252 79 33
P(X = x) 0.05 0.30 0.45 0.14 0.06
1
• Discrete distribution function - example
x – Number of cars 0 1 2 3 4
P(X = x) 0.05 0.30 0.45 0.14 0.06
( )
0 0.05 1 0.30 2 0.45 3 0.14 4 0.06 1.86
xP x
2 2
2 2 2 2 2 2
( )
0 0.05 1 0.3 2 0.45 3 0.14 4 0.06 1.86
0.93
x P x
• Discrete distribution functions
– Binomial distribution
– Poisson distribution
-( ) ( ) (1- )x n x
n xP X x p x C p p
• Continuous random variable
– Random variable that takes on any numerical value
within an interval.
– Possible values of a continuous random variable
are infinite and uncountable.
– Obtained by measurement has a unit of
measurement associated to it.
1/2 + 1/2 = 1
• A continuous random variable has an uncountable
infinite number of values in the interval (a,b)
• The probability that a continuous variable X will assume
any particular value is zero. Why?
• As the number of outcomes increases the probability of each
value decreases.
• This is so because the sum of all the probabilities remains 1.
• When the number of values approaches infinity (because X
is continuous) the probability of each value approaches 0.
The probability of each outcome
1/3 + 1/3 + 1/3 = 1
1/4 + 1/4 + 1/4 + 1/4 = 1
2 outcomes 1/2 + 1/2 = 1
3 outcomes
4 outcomes
• A lot of continuous measurement will
become a smooth curve.
• The probability density curve describe
the probability distribution.
• The density function satisfies the
following conditions:
– The total area under the curve equals 1.
– The probability of a continuous random
variable can be identified as the area under
the curve.
Area = 1
• The probability that x falls between a
and b is found by calculating the area
under the graph of f(x) between a and b.
a b
P(a < X < b)
• Continuous distribution functions
– Normal distribution
Top Related