Question

Let X, Y and Z be independent geometric random variables with the same parameter 0 < p < 1. Let k, n be nonnegative integers. Compute P(X = k|X +Y +Z = n).

Name: let x y and z be independent geometric random variables with the same parameter 0 p 1 let k n be nonnegative integers compute px kx y z n 26872
Uploaded: 2023-03-05T14:18:50-08:00
Duration: 2 min 32 s
Channel: Sri K
Description: let x y and z be independent geometric random variables with the same parameter 0 p 1 let k n be nonnegative integers compute px kx y z n 26872

          Let X, Y and Z be independent geometric random variables with
the same parameter 0 < p < 1. Let k, n be
nonnegative integers. Compute P(X = k|X +Y +Z =
n).

Added by Melinda L.

A First Course in Probability

Sheldon M. Ross 5th Edition

Chapter 7

Instant Answer

Solved by Expert Sri K

03/05/2023

Step 1

We can rewrite this as $Z=n-X-Y$. Substituting this into the joint probability mass function, we get: $P(X=k, Y=m, Z=n-X-Y) = (1-p)^{k-1}p(1-p)^{m-1}p(1-p)^{n-X-Y-1}p$ Now, we want to find the conditional probability $P(X=k|X+Y+Z=n)$. Using the definition of Show more…

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Let X, Y and Z be independent geometric random variables with the same parameter 0 < p < 1. Let k, n be nonnegative integers. Compute P(X = k|X +Y +Z = n).