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MATH/Stat 304
old (but recent) FINAL EXAMINATION
PART I [3 points each]
True/False:
(Assume that all RVs have finite mean and finite variance.)
1.
The authors of our textbook are Richard Scheaffer and Linda Young. ______
2. Suppose that each of f and g is a pdf. Then 1/3 f + 2/3 g is also a pdf.
3. Suppose that each of F and G is a cdf. Then F + G is also a cdf.
4. If E[XY] = E[X] E[Y] then X and Y are independent.
______
______
______
5.
If X and Y are independent RVs, then E[X14Y77] = E[X14 ] E[Y77].
6.
Every cdf is left-continuous but not necessarily right-continuous.
______
______
7. If A, B are events such that P(A) = ¼, P(B) = 1/9, then P(A|B) cannot be
determined.
8.
______
If X and Y are independent RVs then var(X + Y) = var(X) + var(Y).
______
9.
If X and Y are RVs for which the correlation coefficient is not zero, then the
covariance of X and Y must be non-zero as well.
______
10. Suppose that X and Y are exponential RVs with means 4 and 9, respectively,
and that cov(X, Y) = 3. Then E[XY] = 18.
______
11. If A, B and C constitute a partition of a sample space and D is any event, then
P(D) = P(A∩D) + P(B∩D) + P(C∩D).
______
12. If the MGF of a random variable X is given by M(t) =
then E[X] = __________
0.3e t
1  0.7 e t
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PART II [5 points each]
Short answer:
(Assume that all RVs have finite mean and finite variance.)
1.
Which one of the following made no contribution to probability theory? Circle your
choice.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
Jacob Bernoulli
Johann Carl Friedrich Gauss
Jeremy Johns Irons
Siméon Denis Poisson
Augustin Louis Cauchy
Thomas Bayes
Andrei Andreyevich Markov
2. How many arrangements of SKYPE exist for which K and P are not adjacent?
______
3.
In how many ways can 7 indistinguishable rubber ducks be distributed amongst 4
toddlers? (Of course, each toddler is unique.)
______
4. Suppose that X is binomial with parameters n = 10 and p = 0.3. Let Z = 3X. Then
the probability that Z = 8 is
______
5. Given 4 boys and 7 girls; in how many ways can they line up for a photograph such
that no two boys are standing side by side? (Of course, each boy and each girl is
unique.)
______
6. In how many ways can 52 cards be distributed among 4 (distinguishable) players
(each receiving a hand of 13)? (Here the order in which each player receives her 13
cards is not relevant.)
7.
______
In how many ways can Albertine be dealt a hand of 5 cards from a deck of 52 such
that she holds exactly two aces? (Again, the order in which Albertine receives her 5
cards is not relevant.)
______
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8. Let A and B be two events with P(A)=0.27, P(B)=0.41, and P(A∩B)=0.1.
Find the value of P(AC ∩ BC)?
9.
______
Suppose X is a random variable with E(X) = 7 and Var(X) = 11. What is E(X2)?
______
PART III (Discrete RVs) [15 points each]
You may answer all 5 to earn extra credit.
1.
Answer any 4 of the following 5 problems.
Consider a routine screening test for extreme road rage (ERR) disease. Suppose the
frequency of the disease in the population of drivers in Los Angeles (base rate) is
0.5%. The test is highly accurate with a 4% false positive rate and a 13% false
negative rate.
(a) You take the test and it comes back positive. What is the probability that you
have ERR disease?
(b)
You take the test and it comes back negative. What is the probability that
you do not have ERR disease.
2. Two chips are drawn at random and without replacement from an urn that contains five
chips numbered 1 through 5. If the sum of the chips drawn is even, the random variable
X equals 5; if the sum of the chips drawn is odd, X = -3 Find the MGF of X.
3.
Suppose that X is a random variable that assumes values 0, 2 and 3 with probabilities
0.3, 0.1, 0.6 respectively. Let Y = 3(X − 1)2.
(a) What is the support of Y?
(b) What is the expectation of X?
(c) What is the variance of X?
(d) What is the expectation of Y?
(e)
4.
Let FY(y) be the cumulative distribution function of Y. What is FY(7)?
Consider an experiment consisting of two rolls, X and Y, of a fair 4-sided die. Let
B = {min(X, Y) = 2} and let Am = {max(X, Y ) = m} where m may assume values 1, 2, 3, 4.
Find the conditional probability, P(Am| B) for each of m = 1, 2, 3, 4.
5.
Albertine has a bag with 3 coins in it. One of them is a fair coin, and the others are biased trick
coins. When flipped, the three coins come up heads with probability 0.5, 0.7, 0.1 respectively.
Suppose that you choose one of these three coins uniformly at random and flip it three times.
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(a) What is P(HTT)?
(b) Assuming that the three flips, in order, are HTT, what is the probability that the coin that you chose
was the fair coin? (No need to simplify fractions.)
PART IV (Continuous RVs) [15 points each] Answer any 4 of the following 5
problems. You may answer all 5 to earn extra credit.
1. (a) Let A be the parallelogram bounded by the four straight lines:
y = 0, y = x, y = 3, and y = x – 3.
Suppose that
 g ( x, y ) if ( X , Y )  A

f X ,Y ( x, y )  
0 otherwise

(not drawn to scale)
Note: each question may be answered independently of the others.
(a) Express the marginal pdf of Y as integral.
(b) Express the marginal pdf of X as an integral.
(c) Express P(X > 2Y)) as a double integral or sum of double integrals.
2.
Suppose that buses are scheduled to arrive at a bus stop at noon but are always X minutes
late, where X is an exponential random variable with probability density function fX(x) =
(1/5) e−x/5. Suppose that you arrive at the bus stop precisely at noon.
(a) Compute the probability that you have to wait for more than five minutes for the bus to
arrive.
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(b) Suppose that you have already waiting for 10 minutes. Compute the probability that
you have to wait an additional five minutes or more.
3.
In n + m independent Bernoulli (with parameter p) trials, let Sn be the number of successes
in the first n trials and Tm the number of successes in the last m trials.
(a) What is the distribution of Sn? Why?
(b) What is the distribution of Tm? Why?
(c) What is the distribution of Sn + Tm? Why?
(d) Are Sn and Tm independent? Why?
4. Sixty-eight percent of the U.S. population prefer toilet paper to be hung so that the
4. paper dangles over the top; 32% prefer the paper to dangle at the back. In a poll of 500
people in the U.S., the respondents are each asked if they prefer toilet paper to be hung over
the top or to dangle down the back.
(a) What is the probability that at least 65% respond “over the top.”
(a)
(b) What is the probability that at least 34% respond “down the back.”
(b)
5. Suppose that the joint pdf of random variables X and Y is given by:
c ( x  y ) if 0  y  x  2
f X ,Y ( x, y )  
0 otherwise
5.
(a) Find c.
(b) Find P(Y < 0.7 | X < 0.9).
(c) Find the conditional pdf: fY|X(y|x = 1/3).
(d) Find P(Y > X | X = 1/3)
(Added problems: not on old final)
E1. Three tanks fight a three-way duel. Tank A has probability 1/2 of destroying the tank at which it fires, tank
B has probability 1/3 of destroying the tank at which it fires, and tank C has probability 1/6 of destroying the
tank at which it fires. The tanks fire together and each tank fires at the strongest opponent not yet destroyed.
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Form a Markov chain by taking as states the subsets of the set of tanks. Find N, Nc, and NR, and interpret your
results. Hint: Take as states ABC, AC, BC, A, B, C, and none, indicating the tanks that could survive starting in
state ABC. You can omit AB because this state cannot be reached from ABC.
E2.
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The Loveable Eight
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Poisson with parameter . (Expresses the probability of a given number of events occurring in a
fixed interval of time and/or space if these events occur with a known average rate and independently of
the time since the last event. Also: approximates the binomial PMF when n Is large, p is small, and  =
np.)
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Exponential distribution with parameter 
Mean = 1/ 
Variance = 1/ 
Hypergeometric with parameters m and n
Let there be n ways for a “good” selection and m ways for a “bad” selection out of a total of n + m possibilities.
Take a sample of size N.