1. The discrete random variable Y has a probability distribution function P(Y = y) for Y = 1, 2, 3, 4 given by

y
1
2
3
4
P(Y = y)
0.1
0.3
0.4
0.2

Find:

a. E(Y)

b. E(Y2)

c. Use the table to show that E(2Y + 1) = 2E(Y) + 1

d. Use the table to show that VAR(2Y + 1) = 4VAR(Y)

2. The discrete random variable X has a probability distribution function P(X = x) for X = 5, 10, 15 given by

x
5
10
15
P(X = x)
0.2
0.4
0.4

Find:

a. E(X)

b. E(X2)

c. Show that E(3X − 2) = 3E(X) − 2

d. Show that VAR(3X − 2) = 9VAR(X)

3. Given a random variable Z with expected value E(Z) = 3 and variance VAR(Z) = 2, find the following:

a. E(2Z − 3)

b. VAR(2Z − 3)

c. SD(2Z − 3)

4. In a game of DrawBall, two balls are drawn without replacement from a bag containing 4 green balls and 5 red balls. The player receives $12 for each green ball drawn and $6 for each red ball drawn.

(i) Draw a probability tree to show the outcomes of the game.

(ii) If the random variable W = amount won in the game, draw up a probabiity distribution table, and hence find the expected value and variance of W.

(iii) It costs $16 to play a game of DrawBall. If the random variable Z = profit from a game, find the mean and variance of Z.

(iv) Is Drawball a fair game? Explain your answer.

5. X is a random variable with E(X) = 2 and VAR(X) = 3.

a. Find the mean of the random variable 2X + 5.

b. Find the variance of the random variable 2X + 5.