Question

4. (1 point) Consider the following pmf:
x = -2 0 3 5
P(X = x) 0.34 0.07 ? 0.21
What is the

E[
X 1 + X
]
5. (1 point) Screws produced by IKEA will be defective with probability 0.01 independently of each other. What is the expected number of screws to be defective in any given pack? Hint: you should be able to recognize this distribution and use the result from the Exercises at the end of lecture 5.

Problem 2

A fair game, from a probabilistic standpoint, is one in which the cost of playing the game equals the expected winnings of the game. In other words, if X represents our net winnings (winnings - cost of play), a fair game has E[X] = 0. Suppose we pay $6 to play a game in which 5 tacks are thrown into the air. A tack either lands on its side (i) or its head (⊥) with probability 95% and 5% respectively. If no tacks land on their head, we get $0 (i.e. our net winnings is -6 dollars); if exactly one tack lands on its head, we win $10 (i.e. net winnings is $4); if more than one tack lands on its head, we win $20 (i.e. our net winnings is $14).
6. (1 point) Find the expected net winnings.
7. (1 point) Is this a fair game? A. Yes B. No
8. (1 point) What would the cost of the game have to be to make this a fair game? Round your answer to two decimal places (i.e. round your answer to the nearest cent)
Problem 3 If E[X] = 1 and V ar(X) = 5 find:
9. (1 point) E[3 + 1 2X]

10. (1 point) V ar(7−3X)

11. (1 point) E[(4 + X)2] Hint: recall that V ar(X) = E[X2]−(E[X])2

Answer 1