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Chapter 4: Q161S (page 281)

Identify the type of random variable—binomial, Poisson or hypergeometric—described by each of the following probability distributions:

a.p(x)=5xe-5x!;x=0,1,2,...

b.p(x)=(6x)(.2)x(.8)6-x;x=0,1,2,...,6

c.p(x)=10!x!(10-x)!(.9)x(.1)10-x:x=0,1,2,...,10

Short Answer

Expert verified

a. .The given variable is a poisson random variable.

b. .The given variable is a binomial random variable.

c. .The given variable is also a binomial random variable.

Step by step solution

01

Given information

X is a random variable.

02

Identifythe type of random variable when p(x)=.5xe-5x!;x=0,1,2,...

a.

px=.5xe-5x!;x=0,1,2,...=λxe-λx!

whereλ=0.5,x=0,1,2,...

Since the probability distribution of a poisson random variable is

px=λxe-λx!;x=0,1,2,...

Hence, X is a poisson random variable.

03

Identify the type of random variable when p(x)=(6x)(.2)x(.8)6-x;x=0,1,2,...,6

b.

p(x)=(6x)(.2)x(.8)6-x;x=0,1,2,...,6=nxpxqn-x;x=0,1,2,...n

wheren=6,p=0.2,q=0.8,x=0,1,...,6

Since the probability distribution of a binomial random variable is

px=nxpxqn-x;x=0,1,2,...n

Hence, X is a binomial random variable.

04

Identify the type of random variable when p(x)=10!x!(10-x)!(.9)x(.1)10-x:x=0,1,2,...,10 

c.

p(x)=10!x!(10-x)!(.9)x(.1)10-x:x=0,1,2,...,10=n!x!(n-x)!pxqn-x;x=0,1,...,n=(nx)pxqn-x

wheren=10,p=0.9,q=0.1,x=0,1,...,10

Since the probability distribution of a binomial random variable is

px=nxpxqn-x;0,1,2,...,n

Hence, X is a binomial random variable.

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