A sales representative vows to keep knocking on doors until he makes two sales. Given that his probability of success is u, let X= the number of doors he knocks on. Find the probability mass function of X

akuzativo617

akuzativo617

Answered question

2022-11-11

Geometric distribution with multiple successes
"A sales representative vows to keep knocking on doors until he makes two sales. Given that his probability of success is u, let X = the number of doors he knocks on.
Find the probability mass function of X"
My thought is that x cannot be less than 2, since he would have to knock on two doors to make two sales.
I'm thinking the function would be ( x 2 ) ( u 2 ) ( 1 u ) x 2 .
But when I go to find E(x), that doesn't lend itself well to the geometric form I've learned to love.
Am I on the right track at least?

Answer & Explanation

mainzollbtt

mainzollbtt

Beginner2022-11-12Added 13 answers

Step 1
Let us find the probability that X = x, that is, the probability she has to knock on x doors.
Step 2
She has to have 1 success (and therefore x 2 failures) in the first x 1 trials, and then a success. The probability of this is ( x 1 1 ) u ( 1 u ) x 2 u .
Kayley Dickson

Kayley Dickson

Beginner2022-11-13Added 3 answers

Step 1
The negative binomial distribution of trials until the second success is:
P ( X 2 = x ) = ( x 1 1 ) u 2 ( 1 u ) x 2
There must be x 2 failures and one success in any order before the last success.
Step 2
The negative binomial distribution of trials until the r-th success is:
P ( X r = x ) = ( x r + 1 r 1 ) u r ( 1 u ) x r
The geometric distribution is the special case of a negative binomial where r = 1.
P ( X 1 = x ) = ( x 0 ) u 1 ( 1 u ) x 1 = u ( 1 u ) x 1

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