Question

1. Indicate if each of the following is true or false. If false,
provide a counterexample.

(a) The mean of a sample is always the same as the median of
it.

(b) The mean of a population is the same as that of a sample.

(c) If a value appears more than half in a sample, then the mode is equal to the value.

2. (Union and intersection of sets) Let Ω = {1,2,...,6}. Suppose each element is equally likely, i.e., P({ω}) = 1/6 for each ω.

(a) Let A be the set of even numbers in Ω and let B be the set of odd numbers in Ω. Obtain A∪B and A∩B.

(b) Let A ⊆ B. What is the probability of P(A\B)?

(c) Let A be the set of even numbers in Ω. Calculate P(Ω\A).

3. (Permutation and combination) There are 10 landmarks in a city. Ann is supposed to stay in the city for 3 days. Because of time constraint, Ann must choose only one landmark for each day. Suppose Ann does not want to visit the same place more than once. How many options Ann should consider?

4. Ann visits a restaurant and ﬁnds that there are 5 main dishes and 3 desserts. Ann is considering to choose 2 main dishes and 1 dessert. How many options does she have?

5. Bob is supposed to take an exam. There are 5 questions and the event that each question is correctly answered is 30% and those events are independent. Let X be a random variable for the number of correct answers that Bob will make.

(a) Calculate P(X ≥ 1).

(b) Calculate P(X = 2).

(c) Obtain the cumulative distribution function for X.

(d) Calculate the mean and the variance of X.

6. Ann and Bob play a simple game: each independently draws a
number from 1 to 10 each of which is equally likely. The person
with a higher number wins. If they draw a same number, then there
is a draw.

(a) What is the probability of Ann’s winning before drawing a
number?

(b) What is the probability of Ann’s winning conditional on drawing number 7?

(c) Let X1 be a random variable for the number Ann draws; and let X2 be that for Bob. Calculate P(X1 + X2 ≥ 18).

(d) What is the mean of X1 + X2? What is the variance?

7. Ann and Bob are playing a board game. It is almost the end of
the game; Ann is supposed to roll a dice ﬁrst then Bob. Ann wins if
the number on the dice is at least 5, while Bob wins if Ann fails
to win and the number of his dice tuns out to be at least 3. If
both fail then, there will be a draw.

(a) What is the probability of Ann winning the game?

(b) What is the conditional probability of Bob winning the game given that Ann fails to win?

(c) What is the probability of Bob winning the game?

(d) Chris bets on Bob: if Bob wins Chris gets 1, while he loses
2 if Ann wins. A draw gives him 0. What is the mean of the amount
of Chris’ winning?

Answer #1

5. Bob is supposed to take an exam. There are 5 questions and
the event that each question is correctly answered is 30% and those
events are independent. Let X be a random variable for the number
of correct answers that Bob will make.
(a) Calculate P(X ≥ 1).
(b) Calculate P(X = 2).
(c) Obtain the cumulative distribution function for X.
(d) Calculate the mean and the variance of X.
6. Ann and Bob play a simple game: each...

In a game, you roll two fair dice and observe the uppermost face
on each of the die. Let X1 be the number on the first die and X2 be
the number of the second die. Let Y = X1 - X2 denote your winnings
in dollars.
a. Find the probability distribution for Y .
b. Find the expected value for Y .
c. Refer to (b). Based on this result, does this seem like a
game you should play?

Bob and Alice are taking turn shooting arrows at a target, Bob
being the first one to shoot. Alice hits the center of the target
with probability r while Bob hits it with probability p. The first
person to hit the center of the target wins.
(a) What is the probability that Alice wins
(b) Let X be the number of times that Alice shoots an arrow in
the game. Find the PMF of X.
(c) For all k, find...

In a game of “Chuck a luck” a player bets on one of the numbers
1 to 6. Three dice are then rolled and if the number bet by the
player appears i times (where i equals to 1, 2 or 3) the player
then wins i units. On the other hand if the number bet by the
player does not appear on any of the dice the player loses 1 unit.
If x is the players’ winnings in the...

Determine whether the following are binomial experiments. If
not, explain which requirement is violated. If so, state the
distribution of the random variable in the scenario.
A) Bob is playing a game at a carnival where he gets to play
until he loses. The probability that he wins the game is 0.25.
Assume he does not get better each better each time he plays. Let X
count the number of games that Bob wins.
B) Charlie flips a coin three...

Let Ω = {0, 1}^3 , that is, all possible (ordered) triples of
zeros and ones.
Suppose that all outcomes have equal probability.
We define three random variables X1, X2, and X3 on this space
representing the first, second, and third digit, respectively.
We also define X = X1 + X2 + X3.
(i) Compute the values (across Ω) of each of the following
random variables: E(X|X1), E(E(X|X1)|X2), E(X2|X).
(ii) What is the probability mass function of E(X2|X).

A carnival game gives players a 10% chance of winning every
time it is played. A player plays the game 10 times. A. Let X be
the number of times the player wins in 10 plays. What is the most
probable value of X? B. what is the probability that the player
will win at least once?

Suppose you will draw 3 marbles without replacement from each of
four bags. Bag 1 has 3 red marbles and 6 black marbles; bag 2 has
10 red marbles and 20 black marbles; bag 3 has 500 red marbles and
1000 black marbles; and bag 4 has an infinite number of marbles,
but twice as many black marbles as red. Let X1, X2, and X3 be the
numbers of red marbles drawn from bag 1, bag 2, and bag 3,...

Two teams A and B play a series of at most five games. The first
team to win these games win the series. Assume that the outcomes of
the games are independent. Let p be the probability for team A to
win each game. Let x be the number of games needed for A to win.
Let the event Ak ={A wins on the kth trial}, k=3,4,5.
(a) What is P(A wins)?
Express the probability with p and k. Show...

Let X1, X2 be two normal random variables each with population
mean µ and population variance σ2. Let σ12 denote the covariance
between X1 and X2 and let ¯ X denote the sample mean of X1 and X2.
(a) List the condition that needs to be satisﬁed in order for ¯ X
to be an unbiased estimate of µ. (b) [3] As carefully as you can,
without skipping steps, show that both X1 and ¯ X are unbiased
estimators of...

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