Question

Please show and details all work clearly.

Homework 5: Problem 4 Previous Problem List Next (3 points) For x, y = 0 or 1 , let 0, x田y=(1, ifx 0, y = 0 or x = 1,y=1 ifx = 0, y = l or x = 1,y = 0 Let X Bern(0.5) and Y Bern(.5), and assume that X and Y are independent. Find

Answer 1

Here, we are given that both X and Y are independent bernoulli random variables with parameters p = 0.5 each.

Therefore P(X = 0) = P(X = 1) = P(Y = 0 ) = P(Y = 1) = 0.5

a) P(X + Y = 0) = P(X = Y = 0) + P(X = Y = 1) = P(X = 0)P(Y = 0) + P(X = 1)P(Y = 1)

= 0.5*0.5 + 0.5*0.5 = 0.5

Therefore 0.5 is the required probability here.

b) P(X + Y = 1) = 1 - P(X + Y = 0) = 1 - 0.5 = 0.5

Therefore 0.5 is the required probability here.

c) P(X + Y = 0, X =0) = P(X = Y = 0) = 0.5*0.5 = 0.25

Therefore 0.25 is the required probability here.

d) P(X + Y = 0, X =1) = P(X = Y = 1) = 0.5*0.5 = 0.25

Therefore 0.25 is the required probability here.

e) P(X + Y = 1, X =0) = P(X =0)P(Y = 1) = 0.5*0.5 = 0.25

Therefore 0.25 is the required probability here.

f) P(X + Y = 1, X =1) = P(X =1)P(Y = 0) = 0.5*0.5 = 0.25

Therefore 0.25 is the required probability here.