DISCRETE DISTRIBUTIONS OF

RANDOM VARIABLES

3. LESSON

OCW 2020Properties of one-dimensional random variables:

theory and practice

Xabier ErdociaItsaso Leceta

OBJECTIVES

Be able to identify the discrete distribution that the randomvariable is following

After identifying the discrete distribution that a randomvariable is following, be able to calculate different probabilitiesusing probability or distribution function

Have the ability to calculate and interpret the moments ofdifferent discrete distributions

Know the conditions that are necessary to approximate thehypergeometric distribution through the binomial distributionand the binomial distribution through Poisson distribution andbe able to perform properly the approximation

INDEX

3.1. Binary distribution

3.2. Binomial distribution

3.3. Geometric distribution

3.4. Negative binomial distribution

3.5. Hypergeometric distribution

3.6. Poisson distribution

3.1. Binary distribution

3.1. Binary distribution

• When there are only two possible results when performing a single randomtest, the distribution the random variable is following is binary distributionor Bernouilli distribution.

• The two possible results are (Success) or (Failure), being theprobabilities and .

• The distribution that the variable follows is given by:

• The probability function is given by:

• The distribution function is given by:

5

1 1( ) (1 )x x x xp x p p p q

0( ) ( )

1 1

q xF x P X x

x

1X 0X

( 0) 1P X q p ( 1)P X p

Binary( )X p

3.1.

• The mean of a random variable that follows a binary distribution is:

• Knowing the probability function, other moments and measurements canbe calculated. For example, the value of the variance is:

6

( )E X p

2 p q

Binary distribution

3.2.Binomial distribution

3.2. Binomial distribution

• By repeating in n occasions the random test performed in the binarydistribution (being each test independent from the other), the distributionthat follows the random variable which considers the results obtained in allthe tests is a binomial distribution.

• In each test the two possible results are “Success” with probabilityand “Failure” with probability.

• The distribution that the variable follows is given by :

• The probability function is given by:

• The distribution function is given by:

8

( ) (1 ) 0,1,...,x n xn

p x p p x nx

0 0

( ) ( ) ( ) x x

i n i

i i

nF x P X x p i p q

i

B( , )X n p

p

1q p

3.2.

• The mean of a random variable that follows a binomial distribution is:

• Knowing the probability function, other moments and measurements canbe calculated. For example, the value of the variance is:

9

( )E X n p

2 n p q

Binomial distribution

3.3.Geometric distribution

3.3. Geometric distribution

• Taking into account the random test defined in the binary distribution(being each test independent from the other), the distribution that followsthe random variable which considers the number of assays performed untilthe first success (not including the successful test) is geometricdistribution.

• In each test the two possible results are “Success” with probabilityand “Failure” with probability.

• The distribution that the variable follows is given by:

• The probability function is given by:

• The distribution function is given by:

11

( ) 0,1,2,...xp x q p x

0 0

( ) ( ) ( ) x x

i

i i

F x P X x p i q p

G( )X p

p

1q p

3.3. Geometric distribution

• The mean of a random variable that follows a geometric distribution is:

• Knowing the probability function, other moments and measurements canbe calculated. For example, the value of the variance is:

12

( )q

pE X

2

2 q

p

3.4.Negative binomial distribution

3.4.Negative binomial distribution

• Taking into account the random test defined in the binary distribution(being each test independent from the other), the distribution that followsthe random variable which considers the number of tests performed until nsuccesses (not including the n successful test) is geometric distribution

• In each test the two possible results are “Success” with probabilityand “Failure” with probability

• The distribution that the variable follows is given by:

• The probability function is given by:

• The distribution function is given by:

14

1( ) 0,1,2,...x n

n xp x q p x

x

0 0

1( ) ( ) ( )

x xi n

i i

n iF x P X x p i q p

i

BN( , )X n p

p

1q p

3.4.

• The mean of a random variable that follows a negative binomialdistribution is:

• Knowing the probability function, other moments and measurements canbe calculated. For example, the value of the variance is:

15

( )nq

pE X

2

2 nq

p

Negative binomial distribution

3.5.Hypergeometric distribution

3.5.Hypergeometric distribution

• The hypergeometric distribution is similar to the binomial distribution butthe sampling is not independent, that is, observations are made in a finitepopulation without replacement.

• As a finite population, not returning an element to the population afterobserving it, influences the subsequent observation. Proportions do notremain constant as the observations advance.

• In each observation of a population of size there are two possible results,“Success” and “Failure”. In the first observation (test) the probability ofsuccess is and the probability of failure is .

• The distribution that the variable follows is given by:

• The probability function is given by:

17

( ) max(0, ) min( , )

Np Nq

x n xp x n Nq x n Np

N

n

N

H( , , )X N n p

p 1q p

3.5.

• The distribution function is given by:

• The mean of a random variable that follows a negative binomialdistribution is:

• Knowing the probability function, other moments and measurements canbe calculated. For example, the value of the variance is:

18

( )E X n p

2 ( )

( 1)

N n

Nn p q

max(0, ) max(0, )

( ) ( ) ( ) x x

i n Nq i n Nq

Np Nq

i n iF x P X x p i

N

n

Hypergeometric distribution

3.5.

Approximation of hypergeometric distribution through binomial distribution:

• In a random variable that follows an hypergeometric distribution, if thesize of the population, is much larger than the number of observations,this hypergeometric distribution can be approximated through a binomialdistribution.

• Therefore, if ,

• This approach is acceptable when:

19

and 0N n

,n,N

( , , ) ( , )H N n p B n p

10 N n

Hypergeometric distribution

3.6. Poisson distribution

3.6. Poisson distribution

• When an X random variable defines the number of times an event isrepeated in a continuous interval (time, space…), it follows a Poissondistribution. The events occurred in an interval are independent of thoseproduced in another interval, provided that the intervals do not overlap.

• The number of events will always be positive, the parameter will definethe number of events expected in a given interval.

• The distribution that the variable follows is given by:

• The probability function is given by:

• The distribution function is given by:

21

( ) , 0!

xep x x

x

0 0

( ) ( ) ( ) !

ix x

i i

eF x P X x p i

i

( )X P

3.6.

• The mean of a random variable that follows a negative binomialdistribution is:

• Knowing the probability function, other moments and measurements canbe calculated. For example, the value of the variance is:

22

2

( )E X

Poisson distribution

3.6.

Approximation of binomial distribution through Poisson distribution:

• In a random variable that follows a binomial distribution, when thenumber of observations or tests, , is very high and the probability ofsuccess, , is low, this binomial distribution can be approaches throughPoisson distribution.

• Therefore, if ,

• This approach is acceptable when:

23

and 0n p

n

p

( , ) ( )B n p n p P

30 and 0.1n p

Poisson distribution