An advertisement at a Casino claims that the expected value of times a person needs to play a game to win is 3 (the outcome of all games in this exercise is assumed to be independent). Suppose we have reason to believe it takes longer to win. We plan to keep playing the game until we win. If we still haven't won after four plays, we will reject the claim.
a) If the advertisement is true, what is the probability of not winning in the first four games? Namely, what is the chance we are wrong in rejecting the claim?
b) Suppose the true number of times needed to play on average in order to win was truly 5. What is the probability that you will incorrectly accept the claim of the advertisement?
Q2. Suppose that X ~ N(0, 1). Calculate the following two density functions. For each one, be sure to specify the support, i.e., the range of possible values of the random variable.
a) Calculate the density of Y = X^3.
b) Calculate the density of Z = |X|.