math statistics probability Question 3 15 points Let ZGeomet

math statistics, probability Question 3

(15 points) Let Z-Geometric (e). Compute p(5s Z s9).

Solution

Defination of Geometric distribution :

It’s the probability that the first occurrence of success requires k number of independent trials, each with success probability . If the probability of success on each trial is , then the probability that the kth trial (out of k trials) is the first success is

P( Z = k ) = ( 1 - ) ^k-1

for k = 1, 2, 3, ....

Now,

Given problem

Let Z - Geometric () , compute P(5 Z 9)

for this we need to find

P ( 5<=Z<=9) = P(Z=5)+P(Z=6)+P(Z=7)+P(Z=8)+P(Z=9)

Here,

P(Z=5) = ( 1 - ) ^5-1 = ( 1 - ) ⁴ , P(Z=6)= ( 1 - ) ^6-1 = ( 1 - ) ⁵

P(Z=7)=( 1 - ) ^7-1 = ( 1 - ) ⁶ , P(Z=8) = ( 1 - ) ^8-1 = ( 1 - ) ⁷ ,

P(Z=9)= ( 1 - ) ^9-1 =( 1 - ) ⁸

Therefore,

P ( 5<=Z<=9) = P(Z=5)+P(Z=6)+P(Z=7)+P(Z=8)+P(Z=9)

=( 1 - ) ⁴ + ( 1 - ) ⁵ + ( 1 - ) ⁶ + ( 1 - ) ⁷ + ( 1 - ) ⁸