Stat 225 Spring 2010 Exam 2 Review Problems
1.
The discrete random variable X has a PMF described by the table below.
x
2
4
6
8
10
pX(x)
0.2
0.25
0.05
0.3
0.2
(a)
What is the probability that X is between 5 and 9?
(b)
Given that X is at least 4, what is the probability that X is at least 8?
(c)
Calculate the expected value and variance of X.
(d)
Let Z = 10X  5. Find the PMF of Z.
2.
Let Y be a discrete random variable with PMF described by the function below
pY (y) =
𝑦
+3
2
174
if y = 2; 3; 4; 5
0 otherwise
(a)
Verify that Y has a legitimate PMF.
(b)
What is the probability that Y is smaller than 4?
(c)
What is the probability that Y is smaller than 4 if we are told Y is not 2?
(d)
Find the expected value and variance of Y.
3.
The random variable U follows the PMF
pU(u) = k*(5  u)
if u equals 1, 2, 3, 4 or 5
0 otherwise
(a)
Find the value of k
(b)
Find the probability that U is any of 2, 3, or 4.
4.
A special deck of cards consists of two sets of spades and one set of hearts.
A total of 39
cards. The cards are randomly shuffled and the top two are taken off the top of the
deck. The random variable S represents the number of spades among these two cards.
Find the PMF of S.
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5.
Suppose X and Y are random variables with E(X) = 3, E(Y ) = 4 and Var(X)=2. Find
(a)
E(2X + 1)
(b)
E(X  Y )
(c)
E(X
2
)
(d)
E(X
2
 4)
(e)
E((X
–
4)
2
)
(f)
Var(2X  4)
6.
Samantha plans to attend a volleyball game and wants to get some of her friends to go
with her. Let X represent the number of the seven friends she calls that are interested.
The probability any one of them will say 'yes' is 0.8, regardless of the responses from the
others. Answer the following questions.
(a)
What is the distribution of X and what are its parameter(s)?
(b)
What is the probability that at least four of her friends say 'yes'?
7.
Identify the parameters p and n for each of the following Binomial distributions and find
the expected value and variance of the random variable described:
(a)
The number of heads in 5 tosses of a fair coin.
(b)
The number of correct answers on a multiple choice test if each of the 25
questions has 5 possible answers and the student guesses randomly.
(c)
The number of 6's in 100 tosses of a fair dice.
8.
A pair of fair dice is rolled 10 times. Let X be the number of (double) rolls in which we
see at least one six.
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 Spring '08
 MARTIN
 Probability

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