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.