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