A study examined whether there was correlation between how much football a person watched and how bald the person was. The time T watching football was measured on a 0, 1. 2 scale such that T = 0 if a person never watched football, T = 1 if a person watched football occasionally, and T = 2 if a person watched a lot of football. Similarly, baldness B was measured on the Same scale: B = 0 for a person with a full head of hair, B -- 1 for a person with thinking hair, and B = 2 for a person who has not much hair at all. The experiment was to learn B and T for a randomly chosen person, equally likely to be a umm (event Al) or a woman (event lU). The study found that given a person was a man (event M), random variables B and T were conditionally independent.. Similarly, given that a person was a woman (event W), B and T were conditionally independent. Moreover, B and T ha(l conditional joint PMFs

(a) Find the conditional PMF PB,T(b. t) of B and T given that. a person is a woman.
(b) Find 11W conditional PMF PB,T|M( b,t) of B and T given that a person is a man.
(c) Find the joint PMF PB,T(b, t).
(d) Find the covariance of B and T. Are B and T independent?

PBM(b) 0.2 0.3 0.5 rit) 0.2 0.2 0.6 Pw(b) 0.6 0.3 0.1 Priw(t) 0.6 0.2 0.2 BIW