5. You and your friend are playing a game. You start by selecting a number X uniformly at random from [0, 1]. Your friend picks numbers Y, Y2, ... uniformly in [0, 1] until they pick a number larger than X/2. (a) (10 points) Find the expected number of times, N, your friend needs to pick a number. Hint: Condition on X = X. (b) (10 points) Find the expected sum of your friend's numbers given that they had to pick N numbers. (c) (5 points) Find the expected sum of your friend's numbers. 5. You and your friend are playing a game. You start by selecting a number X uniformly at random from [0, 1]. Your friend picks numbers Y, Y2, ... uniformly in [0, 1] until they pick a number larger than X/2. (a) (10 points) Find the expected number of times, N, your friend needs to pick a number. Hint: Condition on X = X. (b) (10 points) Find the expected sum of your friend's numbers given that they had to pick N numbers. (c) (5 points) Find the expected sum of your friend's numbers