Assume we flip a fair coin twice, independently of each other. Define the following random variables:
X = the number of heads,
Y = the number of tails,
Z = the number of heads times the number of tails.
(a) Determine the expected values of these three random variables.
(b) Are X and Y independent random variables?
(c) Are X and Z independent random variables?
(d) Are Y and Z independent random variables?

The vice president of programming at nbc is finalizing the prime-time schedule for the fall. she has decided to include a hospital drama but is unsure which of two possibilities to select. she has a pilot called the surgeon and another called critical care. to her make a final decision, a sample of 20 viewers from throughout the united states was asked to watch the two pilots and indicate which show they prefer. the results were that 12 liked the surgeon, 7 liked critical care, and one had no preference. state the decision rule, using the 0.10 significance level. h0: π = 0.50 h1: π ≠ 0.50 1. reject h0 if number of + signs is either < = or > = 2. what is your decision regarding h0? is there a preference for one of the two shows?