Solution to Question 1 from UoL exam 2017, Zone B (FN3142)
This is a relatively simple problem but surprisingly many students cannot answer it. I provide two answers, the first of which gives a general idea about how to solve such problems.
Imagine the following gamble. First, flip a fair coin to determine the amount of your bet: if heads, you bet $1, if tails you bet $2. Second, flip again: if heads, you win the amount of your bet, if tails, you lose it. For example, if you flip heads and then tails, you lose $1; if you flip tails and then heads you win $2. Let be the amount you bet, and let be your net winnings (negative if you lost).
(a) [5 Marks] Show that the covariance between X and Y is zero.
(b) [5 Marks] Show that X and Y are not independent.
(a) When dealing with discrete random variables, a probability table is the best tool. is described by the table
Table 1. Values and probabilities for
For , the two bets have equal probabilities of 1/2, so the mean is
Table 2. Values and probabilities for
For , the possible outcomes are 1,-1,2,-2, and each of them has probability 1/4, so the mean is
To find the covariance, we use the shortcut
For a pair of variables we need a two-way table
Table 3. Values and probabilities for the pair
In the main body of the table we have probabilities of joint outcomes . The usual mean formula applies and gives
(b) For independent variables, all joint probabilities equal products of individual (marginal) probabilities. From Table 3 we see that this is not true, so the variables are not independent.
For this method, we still need everything that led to equation (1) but don't need Table 3. Denote a function which is 1 on the set and zero outside it. Since can be either 1 or 2, we have
identically on the sample space. Hence,
(replacing 1 by the sum (2))
(pulling out constant values of )
because on the sets the outcomes "lose" and "win" have equal probabilities.