Question

The joint probability mass function (PMF) of two random variables x and y is given as:

0.01, x=1,2,...,10, y= 1,2,...,10
Pₓᵧ(x,y) = { 0, Otherwise
a. find the PMF of W = min(X,Y)
b. Let A be the event of min(X,Y) > 5. Find the conditional PMF Pₓᵧ (x,y).

Answer 1

The PMF of W, the minimum of X and Y, can be determined by considering the joint probabilities of X and Y. The conditional PMF PXᵧ(x,y) given the event A: min(X,Y) > 5, can be calculated by considering the joint probabilities for the values that satisfy A.

Step-by-step explanation:

To find the PMF of W = min(X,Y), we need to consider all possible values of W. Since W represents the minimum of X and Y, it can take any value between 1 and 10, inclusive. For each value of W, we calculate its probability by considering the joint probabilities of X and Y. The PMF of W is:

WPW(W)

1) 0.01

2) 0.02

3) 0.03

4) 0.04

5) 0.05

6) 0.06

7) 0.07

8) 0.08

9) 0.09

10) 0.1

To find the conditional PMF PXᵧ(x,y) given the event A: min(X,Y) > 5, we need to consider the joint probabilities of X and Y for all values that satisfy A. Since A is true whenever both X and Y are greater than 5, the conditional PMF PXᵧ(x,y) is:

xyPXᵧ|A(x,y)

6 6 0.064516129

7 7 0.072727273

8 8 0.081632653

9 9 0.090909091

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