Geoff is running a carnival game. He has 15 marbles in a bag: there are 4 green marbles, 7 red marbles and 4 yellow marbles. To play a round of the game, a player randomly takes out a marble from the bag, notes the color and replaces it, then pulls a second marble from the bag and notes the color. So in effect, the player pulls 2 marbles from the bag. (However, the first marble is put back in the bag and so potentially could be pulled twice.) Green marbles win 5 points, red marbles win 2 point and yellow marbles lose 2 points.

Let X be the random variable that describes the number of points won by a player playing a single round of Geoff's marble game. Find the probability distribution for X. Give values for X as whole numbers and probabilities as decimal values to 3 decimal places. Enter the values for X in ascending order (lowest to highest) from left to right in the table.

X _____ _____ _____ _____ _____ _____

P(X=x) _____ _____ _____ _____ _____ _____

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Suppose Y is a discrete random variable with probability distribution given by:

P(Y = k) = k/21 , for k = 1, 2, 3, 4, 5, 6

a)Calculate the expected value of Y. Give your answer to 2 decimal places.

E[Y] =

b)Calculate the variance of Y. Give your answer as a decimal to 2 decimal places.

VAR[Y] =