Suppose X and Y are discrete rvs such that PX Y 1 This me

Suppose X and Y are discrete r.v.s such that P(X = Y ) = 1. This means thatX and Y always take on the same value.
(a) Do X and Y have the same PMF?
(b) Is it possible for X and Y to be independent?

Solution

P(X=y) = 1

This means x,y takes values from 0 to 1 under the line y =x and above x and y axis

Hence x varies from o to y and y varies from 0 to rt 2.

(a) PMF of x is = rt 2 and Pmf of y is = y

b) P(X,y) not equals y and hence x and y cannot be independent.