Question

2) Consider the sample space of three coin tosses: Ω = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT }.

Assuming all elements to be equally likely, we assign P({ωi}) = 1/8, i = 1, 2, 3, 4, 5, 6, 7, 8. Define random variable to capture the second and third outcomes of the toss:

X2 = { 0, if second outcome is T; 1, if second outcome is H

and X3 = { 0, if third outcome is T; 1, if third outcome is H

Also define a random variable to count the number of Heads: Y = number of Heads.

Solve the following problems.

A) Express X2, X3, and Y as functions of ω, as done in class.

B) Find the pmfs of X2, X3, and Y

C) Find the joint pmf of X2 and Y . Are X2 and Y independent?

D) Find the joint pmf of X2 and X3. Are X2 and X3 independent?

E) Find the expected values of X2, X3, and Y .
MUST SHOW ALL WORK FOR FULL CREDIT!!!

Answer 1