Download PROBABILITY FINAL (1) Let X be a random variable uniformly

PROBABILITY FINAL
(1) Let X be a random variable uniformly distributed on the set {−1, 0, 1}}. Find the variance of X.
(a) 1/3
(b) 2/3
(c) 3/3
(d) 1/6
(e) 5/6
(2) What is the expected number of times one must flip a fair coin before one observes at least one H and one
T ? In other words, one must observe both a H and a T .
(a) 1
(b) 2
(c) 3
(d) 4
(e) 5
(3) Mary and Jane play a matching pennies game with a fair coin. They simulateously flip a penny. If the
coins match (both H or both T ) Mary wins a penny, and if they do not match, Jane wins a penny. If they
play four rounds, what is the probability that Mary ends up with a greater number of pennies than Jane
given that Mary wins the first round.
(a) 1/8
(b) 2/8
(c) 3/8
(d) 4/8
(e) 5/8
(4) Suppose that in a certain class 40% of the students are engineering majors, 30% are math majors, and 10%
double major in both math and engineering. What percentage of students are neither math nor engineering
majors?
(a) 10%
(b) 15%
(c) 25%
(d) 30%
(e) 40%
(5) Suppose X and Y are independent and uniformly distributed on {−1, 0, 1}. What is the probability that
the sum of X and Y equals 0?
(a) 1/9
(b) 2/9
(c) 1/2
(d) 1/4
(e) 1/3
(6) Suppose X is uniformly distributed on [0, 1] and Y is uniformly distributed on [−2, 0]. If X and Y are
independent, what is the probability that X + Y is non-negative?
(a) 1/8
(b) 1/6
(c) 1/4
(d) 1/3
(e) 1/2
(7) Let X be a Poisson random variable with mean of 1. For what value of k is the probability that X ≤ k
approximately equal to .98 ?
(a) 1
(b) 2
(c) 3
(d) 4
(e) 5
PROBABILITY FINAL
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(8) Urn A contains 3 red and 4 blue balls. Urn B contains 2 red and 2 blue balls. Mary flips a fair coin. If it
lands H she selects a ball from urn A, and if it land T she selects a ball from urn B. Suppose she ends up
with a red ball. What is the probability the coin landed H?
(a) 6/13
(b) 7/13
(c) 8/13
(d) 9/13
(e) 10/13
The next three problems involve the Poisson distribution. Note that the variance of a Poisson distribution is equal to the expected value, and the sum of independent Poisson random variables is
Poisson.
(9) The number of typographic errors on each page of a 500 page novel can be modeled as a Poisson random
variable. (Assume here and in the following two problems that the number of errors of different pages are
independent.) Suppose the expected number of errors on a page is .2 . What is the probability that the
novel contains no typographic errors?
(a) e−500
(b) e−400
(c) e−300
(d) e−200
(e) e−100
(10) The number of typographic errors on each page of a 500 page novel can be modeled as a Poisson random
variable. Suppose the expected number of errors on a page is .2 . Recall that Chebyshev’s inequality says
that
Prob(|X − µ| > ε) ≤
var(X)
.
ε2
Which of the following statements can be justified by using Chebyshev’s inequality?
(a) The probability that the total number of errors in the novel is between 90 and 110 is less than 1/2.
(b) The probability that the total number of errors in the novel is greater than 110 is less than 1/2.
(c) The probability that the number of errors in the novel is less than 110 is greater than 1/2.
(d) The probabiltiy that the number of errors in the novel is between 80 and 120 is less than 1/4.
(e) The probability that the number of errors in the novel is between 80 and 120 is greater than 3/4.
(11) The number of typographic errors on each page of a 500 page novel can be modeled as a Poisson random
variable. Suppose the expected number of errors on a page is .2 . Use the Central Limit Theorem to
estimate the probability that the novel contains between 90 and 100 errors.
(a) .34
(b) .39
(c) .41
(d) .45
(e) .29
(12) The life span of an electronic component that costs $100 is exponentially distributed with a mean on 10
years. The manufacturer will refund $100 if it fails within the first year. If it fails after one year but before
two years the manufacturer will refund $50. What is the expected refund cost to the manufacturer on the
component?
(a) 100 − 50 e−(1/10) − e−(2/10)
(b) 100 − 50 e−(1/10) + e−(2/10)
(c) 100 + 50 e−(1/10) − e−(2/10)
(d) 50 + 50 −e−(1/10) + e−(2/10)
(e) 50 − 50 e−(1/10) − e−(2/10)
(13) A manufacturer must decide on the guarantee policy for an electronic component whose lifespan is exponentially distributed. The guarantee will provide a full replacement if the device fails in the first T years.
If the expected lifespan is 10 years , how should the manufacturer select T if they want to be 95% certain
that they will not have to replace the component.
(a) T = ln(100/95)
(b) T = 5ln(100/95)
(c) T = 10ln(100/95)
(d) T = 15ln(100/95)
(e) T = 20ln(100/95)
PROBABILITY FINAL
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(14) Suppose you select, without replacement, 5 cards from a standard 52 card deck. What is the probability
that you select at least one ace?
(a) [(52)(51)(50)(49) − (47)(46)(45)(44)] /(52)(51)(50)(49)
(b) (47)(46)(45)(44)/(52)(51)(50)(49)
5
(c) (12/13)
5
(d) 1 − (12/13)
52
(e)
(1/13)(12/13)4
5
(15) Suppose the random variable X is continuously distributed on [0, 1] with density f (x) proportional to x3 .
What is the expected value of X?
(a) 1/5
(b) 2/5
(c) 3/5
(d) 4/5
(e) 1
(16) Suppose X and Y are jointly distributed on the unit square, [0, 1] × [0, 1] with joint density given by
f (x, y) =
kx if y < x
1 if y ≥ x
where k is a constant. What is the probability that Y < X?
(a) 1/4
(b) 1/3
(c) 2/5
(d) 1/2
(e) 3/5
(17) Suppose X and Y are jointly distributed and take values in the set {0, 1}. The probability that both X
and Y are 0 is 1/3. The probability that they are both 1 is 1/4. The probability that X = 0 and Y = 1 is
the same as the probability that Y = 0 and X = 1. What is the probability that X = 0?
(a) 10/24
(b) 11/24
(c) 12/24
(d) 13/24
(e) 14/24
(18) Suppose X is uniformly distributed on [0, 1]. What is the probability that 1/2 < X < 2/3 given that
X > 1/4?
(a) 1/6
(b) 2/9
(c) 1/3
(d) 4/9
(e) 5/9
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(19) You have two coins; one is fair and the other lands H with probability .9. You will win $1 if you can
correctly guess which coin is fair. Suppose you pay $.10 (ten cents) to make a “test flip” before you guess.
What is your expected payoff without the test flip and with the test flip (assuming you pay ten cents for
the test flip)?
(20) Let X` be independent random variables with mean 2 and variance 1. Let Wn and Yn be defined by
X1 + X2 + ... + Xn − 2n
Wn =
n
and
X1 + X2 + ... + Xn − 2n
√
Yn =
n
Find the limits,
limn→∞ var(Wn )
and
limn→∞ var(Yn )
You must provide some explanation for your work. A number alone will not suffice.