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

$ (prize)
$(profit)
x = # of
heads
P(x)
$1
$3
0
0.03125
$ 1
$3
1
0.15625
$1
$ 1
2
0.31250
$1
$1
3
0.31250
$ 1
$0
4
0.15625
$ 1
$0
5
0.03125
1. A fair coin is tossed five times. The probability of
observing “x” heads among the 5 coins is given in the table above.
A particular game consists of tossing a fair coin 5 times. It costs
$1 to play the game; you...

A spinner game has a wheel with the numbers 1 through 30 marked
in equally spaced slots. You pay $1 to play the game. You pick a
number from 1 to 30. If the spinner lands on your number, you win
$25. Otherwise, you win nothing. Find the expected net winnings for
this game. (Round your answer to two decimal places.)
A game costs $1 to play. A fair 5-sided die is rolled. If you
roll an even number, you...

1. Let X be the number of heads in 4 tosses of a fair coin.
(a) What is the probability distribution of X? Please show how
probability is calculated.
(b) What are the mean and variance of X?
(c) Consider a game where you win $5 for every head but lose $3
for every tail that appears in 4 tosses of a fair coin. Let the
variable Y denote the winnings from this game. Formulate the
probability distribution of Y...

Suppose you play a $2 scratch off lottery game where there is a
1 in 4 chance to win $2, a 1 in 10 chance to win $10, a 1 in 15
chance to win $250, and a 1 in 35 chance to win $5000.
Construct a probability distribution for x where x represents
the possible net winnings, i.e., winnings minus cost, including the
scratch off lottery games that yield no winnings.
Use the probability distribution to calculate the expected...

Two balls are chosen randomly from an urn containing 9 yellow, 5
blue, and 3 magenta balls. Suppose that we win $3 for each blue
ball selected, we lose $2 for each yellow ball selected and we win
$0 for each magenta ball selected. Let X denote our winnings. What
are the possible values of X, what are the probabilities associated
with each value (i.e., find the probability mass function of X),
and what is the expectation value of X,E[X]?

Given a random variable X has the following pmf:
X
-1
0
1
P[X]
0.25
0.5
0.25
Define Y = X2 & W= Y+2.
Which one of the following statements is not true?
A) V[Y] = 0.25.
B) E[XY] = 0.
C) E[X3] = 0.
D) E[X+2] = 2.
E) E[Y+2] = 2.5.
F) E[W+2] = 4.5.
G) V[X+2] = 0.5.
H) V[W+2] = 0.25.
I) P[W=1] = 0.5
J) X and W are not independent.

Penalty Worksheet – Due
10-19 Expected
Value of a
Game Name_______________________
Instructions:You must show your work and all
work must be organized and easy to follow. You will have to make an
appointment with me to discuss your work. You will not receive
credit if you cannot adequately explain your work to me in
person.
I pick a digit (an integer from 0 to 9, inclusive). Then, for a
dollar, you get to pick three digits randomly, with replacement, by
spinning a...

Consider the following game. You flip an unfair coin, with P(H)
= 1/4 and P(T) = 3/4, 100 times. Every time you flip a heads you
win $8, and every time you flip a tails you lose $3. Let X be the
amount of money you win/lose during the game. Justify your answers
and show all work. Compute E(X) andCompute V (X).

4. Given x: -3 / 0 / 3 P(X =x): .5 / .2 / .3 Find the variance
given that the expected value is - .6
A. 4.1 B. 2.85 C. 8.142 D. 2.02 E. 6.84
5. A probability distribution has a mean of 10 and standard
deviation of 1.5. Use Chevbychev’s Inequality to estimate the
probability that an outcome will be between 7 and 13.
A. 4 B. 2 C. 5 D. 4

1. Find P{X = x} for x = 0, 1, 2, 3, 4, 5 for a Bin(5, 3/7)
random variable.
2. Find the first and third quartiles as well as the median for
a Beta(3, 3) random variables.
3. Find P{X ≤ x} for x = 0, 1, 2, 3 for a χ 2 2 random
variable.
4. Simulate 80 Pois(5) random variable. Find the mean and
variance of these simulated values.

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