Question 2 (25%)
Toss a fair coin for five times, and let discrete random variable X be the difference between
the number of heads coming up and the number of tails coming up. That is, if let discrete random
variable Z be the number of heads coming up, and discrete random variable W be the number of
tails coming up, then $X = Z - W$.
i. Write down the entire probability distribution function of X (preferably as a table). Hint:
Z and W have perfect linear relationship, and hence X can be written as a function of only Z or
only W.
ii. What is the expected value of X?
iii. What is the variance of X?
Now consider the same random experiment described above, and let discrete random variable
Y be the indicator of whether the head coming up in the first tossing. That is, Y = 1 if the head
coming up in the first tossing, and Y = 0 if the tail coming up in the first tossing.
iv. Write down the entire conditional probability distribution function of discrete random vari-
able X, given that Y = 1 (preferably as a table).
v. Without calculating the joint probability distribution function of X and Y, guess whether
the covariance between X and Y will be positive, negative, or zero. Provide your intuition here.