You have 8 pairs of mittens, each a different color. Left and right mittens are distinct. Suppose that you are fostering (possibly imaginary) kittens, and you leave them alone for a few hours with your mittens. When you return, you discover that they have hidden 4 mittens! Suppose that your kittens are equally likely to hide any 4 of your 16 distinct mittens. Let X be the number of complete, distinct pairs of mittens that you have left. (a) Compute the probability mass function of X, px(k). (Hint: You may want to first figure out the range Ωx).
(b) Find E[X] using your answer to part (a) and the definition of expectation.
(c) Find E[X] again, but using linearity of expectation this time and indicator RVs. Are these indicator RVs independent? Rhetorical (answering this next part is optional): Which method was easier? What if I increased the number of pairs of mittens?