Question

Suppose a game is designed as follows. Each player that enters the game is asked to draw a ball from a bag that contains 10 balls: 4 red balls, 3 blue balls and 3 green balls. Assume that the bag is not transparent so that the players cannot see the balls inside the bag when drawing. Then the player wins $5 if he draws a blue ball; the player wins $3 if he draws a red ball; the player loses $10 if he draws a green ball. Here we define the profit as the dollar amount the player wins or loses. So the profit is positive if the player wins money and the profit is negative if the player loses money. (a) Find the moment generating function of the profit for a randomly chosen player. Show your steps. (b) Using the moment generating function that you found in part (a), find the expectation of the profit for a randomly chosen player. (c) Suppose 10 players are randomly chosen to play the game, and assume that the outcomes of the 10 players are independent of each other. Find the moment generating function of the total profit made by the 10 chosen players.

Answer 1