July 4, 2022
Probability Theory: Homework problems
Homework 1.
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1. A “traditional” three-digit telephone area code is constructed as follows. The first digit is from
the set {2, 3, 4, 5, 6, 7, 8, 9}, the second is either 0 or 1, the last is from the set {1, 2, 3, 4, 5, 6, 7, 8, 9}. (a)
How many area codes like this are possible? (b) How many such area codes start with 5?
2. In how many ways can three novels, two mathematics books and one chemistry book be arranged
on a shelf if (a) any arrangement is allowed; (b) math books must be together and the novels must be
together; (c) only the novels must be together.
3. Verify the odds of various events in the California State Lottery game MEGA Millions, as shown
https://www.calottery.com/draw-games/mega-millions#section-content-3-3
4. (I) Seven different gifts are distributed among 10 (different) children. How many different outcomes
are possible if every child can receive (a) at most one gift; (b) at most two gifts; (c) any number of gifts
(II) Answer the same questions if the gifts are identical (but children are still different).
5. (I) Twenty different gifts are to be distributed among 7 (different) children. How many outcomes
are possible if every child is to receive (a) at least one gift (b) at least two gifts (c) any number of gifts.
(II) Answer the same questions if the gifts are identical (but children are still different).
(III) Now try to generalize problems 4 and 5.
6. Consider the two-dimensional Cartesian (standard rectangular x, y) coordinate system on the
plane. You move around the points with integer coordinates in such a way that, at each step you can
go either one unit up or one unit to the right (that is, from (0, 0) you can go to either (1, 0) or (0, 1);
from (2, 3) you can go either to (3, 3) or (2, 4), etc.) Count the number of different paths from the point
(0, 0) to (a) the point (4, 3); (b) the point (4, 3), if you have to visit the point (2, 2); (c) the point (4, 4),
if you are not allowed to go above the line x = y (but you are allowed to hit the line, e.g. by visiting
point the (1, 1)).
7. You have $20K to invest, and have a choice of stocks, bonds, mutual funds, or a CD. Investments
must be made in multiples of $1K, and there are minimal amounts to be invested: $2K in stocks, $2K
in bonds, $3K in mutual funds, and $4K in the CD. Count the number of choices in each situation: (a)
You want to invest in all four; (b) You want to invest in at least three out of four.
8. Two dice are rolled. Introduce the following events:
(1) E : “the sum is odd”
(2) F : “at least one number is 1”
(3) G : “the sum is 5”
List the elementary outcomes in each of the following events: EF, E
F, FG, EF c, EFG. For this
problem, would you care whether the dice are fair?
Homework 2.
1. At a certain school, 60% of the students wear neither a ring nor a necklace, 20% wear a ring,
30% wear a necklace. Compute the probability that a randomly selected student wears (a) a ring OR
a necklace; (b) a ring AND a necklace.
2. A school offers three language classes: Spanish (S), French (F), and German (G). There are 100
students total, of which 28 take S, 26 take F, 16 take G, 12 take both S and F, 4 take both S and G, 6
take both F and G, and 2 take all three languages.
(1) Compute the probability that a randomly selected student (a) is not taking any of the three
language classes; (b) takes EXACTLY one of the three language classes.
(2) Compute the probability that, of two randomly selected students, at least one takes a language
3. Two fair dice are rolled. Compute the probability that the number on the first is smaller that the
number on the second.
4. Alice and Bob are in a group of N people who are randomly arranged (a) in a row (b) in a circle.
In each case, compute the probability that Alice and Bob are next to each other.
5. Two fair dice are rolled. For i = 2, 3, . . . , 12, compute the probability that the first one shows 6
given that the sum is i. (Of course, it is zero if i < 7).
6. (I) Let A and B be two events such that P (A) = 0.5 and P (B) = 0.6. (a) Can A and B be
mutually exclusive? (b) Assuming that A and B are independent, compute P (A
(II) Let A, B, C be three events such that P (A) = 0.5, P (B) = 0.6, P (C) = 0.8 (a) Can any
two of these events be mutually exclusive? (b) Assuming that the events are independent, compute
(III) Let A and B be events such that P (A) = 0.7 and P (B) = 0.8.
(a) Circle the possible values of P (A
B): 0.3 0.5 0.8 0.9
(b) Circle the possible values of P (A
B): 0.7 0.8 1
You need to explain each of your conclusions. For example, if you think that P (A
B) can be 0.5,
you can draw the corresponding Venn diagram, and if you think that P (A
B) cannot be 1, you can
support your claim with suitable formulas.
7. True or false: if A and B are events such that 0 < P (A) < 1 and P (B|A) = P (B|Ac), then A and
B are independent?
8. The probability that a student passes the first midterm is p, the probability that the student passes
the second midterm is q, and the probability that the student passes at least one of the midterms is r.
Identify the constraints on the possible values of p, q, r, and then compute the probability s, in terms
of p, q, r, that the student passes both midterms. [0 ≤ p, q, r ≤ 1, p + q ≥ r; by inclusion/exclusion,
p+ q − s = r, so 0 ≤ s = p+ q − r ≤ 1]
Homework 3.
1. In a certain community, 36% of all the families have a dog and 30% have a cat. Of those families
with a dog, 22% also have a cat. Compute the probability that a randomly selected family (a) has both
a dog and a cat; (b) has a dog GIVEN that it has a cat.
2. Three fair dice have different colors: red, blue, and yellow. These three dice are rolled and the face
value of each is recorded as R,B, Y , respectively. Compute the probability that B < Y < R. You can
proceed as follows: (a) compute the probability that no two dice show the same number; (b) compute
the probability that B < Y < R given that all the numbers are different; (c) solve the problem.
3. Suppose that 5% of cars and 0.25% of trucks are yellow. Compute the probability that a randomly
selected yellow vehicle is a car if (a) the proportion of cars and trucks in the population is that same;
(b) there are twice as many trucks as cars in the population.
4. English and American spellings are rigour and rigor, respectively. At a certain hotel, 40% of
guests are from England and the rest are from America. A guest at the hotel writes the word (as either
rigour or rigor), and a randomly selected letter from the spelling turns out to be a vowel. Compute the
probability that the guest is from England.
5. Two people, A and B, are involved in a duel. The rules are as follows: shoot at each other once;
if at least one is hit, the duel is over, if both miss, repeat (go to the next round), and so on. Denote by
pA and pB the probabilities that A hits B and B hits A with one shot, and assume that hitting/missing
is independent from round to round. Compute the probabilities of the following events: (a) the duel
Figure 1. A random connection
ends and A is not hit; (b) the duel ends and both are hit; (c) the duel ends after round number n; (d)
the duel ends after round number n GIVEN that A is not hit; (e) the duel ends after n rounds GIVEN
that both are hit; (f) the duel goes on for ever.
6. On Figure 1, each of the five connections can be open or closed independently of other connections.
The probability to have a specific connection closed is p.
(a) Compute the probability that there is a path of closed connections from A to C. [One possible
solution: by inclusion-exclusion. To keep track of what you are doing, it might actually be easier to
denote the probability of each connection by a different letter].
(b) Compute the conditional probability that the connection along the diagonal BD is closed given
that there is a path of closed connections from A to C.
Whatever answer you get in both parts (a) and (b), check that the result is a function that is
monotonically increasing from 0 when p = 0 to 1 when p = 1 (it might be easier to do it using a
computer algebra system).
7. Thirty percent of drivers stopped by police are drunk. The sobriety test is 95% accurate on drunk
drivers and 80% accurate on sober drivers. A driver is stopped and fails the sobriety test. (a) What is
the probability that the driver is drunk? (b) How many times should the sobriety test be administered
to be 99.999% sure that the drive is drunk? Assume that, when the sobriety test is administered several
times, the test results are independent.
Homework 4.
1. Five USC students and five UCLA students are ranked according to their performance on a test.
Assume that no two test scores are the same and all possible rankings are equally likely. Let X be the
highest ranking of a USC student. Compute the distribution of X.
2. A coin is tossed n times. Let X be the difference between the number of heads and the number
of tails. Compute the possible values of X. Do we care whether the coin is fair or not?
3. A fair coin is tossed n times. Let X be the difference between the number of heads and the number
of tails. Compute the distribution of X when (a) n = 3; (b) n = 4; (c) n ≥ 2 is arbitrary. How is the
distribution of X different from the binomial distribution?
4. Consider the following strategy for paying the roulette. Bet $1 on red. If red appears (which
happens with probability 18/38), then take the $1 profit and stop playing for the day. If red does
not appear, then bet additional $1 on red each of the following two rounds, and then stop playing for
the day no matter the outcome. Let X be the net gain/loss. (a) Describe the distribution of X; (b)
Compute P (X > 0); (c) Compute the expected value of X; (d) Would you consider this a winning
strategy? Explain your reasoning.
5. Two teams play a series of games until one of the teams wins n games. In every game, both teams
have equal chances of winning and there are no draws. Compute the expected number of the games
played when (a) n = 2; (b) n = 3. (To keep track of what you are doing, it can be easier to use different
letters for the probabilities of win for the two teams).
6. A communication channel transmits a signal as sequence of digits 0 and 1. The probability of
incorrect reception of each digit is p. To reduce the probability of error at reception, 0 is transmitted as
00000 (five zeroes) and 1, as 11111. Assume that the digits are received independently and the majority
decoding is used. Compute the probability of receiving the signal incorrectly if the original signal is (a)
0; (b) 101. Evaluate the probabilities when p = 0.2.
7. In a certain jurisdiction, it takes at least 9 votes of a 12-member jury to get a conviction. Assume
(1) 65% of all defendants are guilty;
(2) the probability that a juror will convict an innocent is 0.1;
(3) the probability that a juror will acquit a guilty is 0.2;
(4) each juror votes independently of the rest of the panel;
Compute the probabilities of the following events: (a) the panel renders a correct decision; (b) the
defendant is convicted. Use your favorite software package to evaluate the probabilities numerically.
Homework 5.
1. On a certain highway, there are, on average, 2.2 cars abandoned every week. Assuming Poisson
distribution, compute the probability that (a) there will be no cars abandoned next week; (b) there will
be at least 5 cars abandoned next month.
2. Suppose that the number of times a person catches cold in a year is a Poisson random variable
with parameter λ = 5. A new drug is claimed to reduce this parameter λ to 3 for 75% of the population
and has no effect on the rest of the population. Somebody takes the drug and gets two colds in a year.
Compute the probability that the drug was beneficial for that person. (Note: this is a classical Bayes
rule problem).
3. At time t0 = 0, a fair coin is tossed and lands heads. Then, at a random time T > 0, the coin is
tossed again. Given a t > 0, compute the probability that the coin shows heads at time t, if T is the
moment of the first event in a Poisson process with parameter λ. How will the answer change if the
coin is not fair?
4. An urn contains four black and four white balls. Four balls are taken out of the urn. If two are
black and two are white, the experiment ends. Otherwise, the balls are returned to the urn and the
experiment is repeated. Denote by X the number of experiments conducted. Decribe the probability
distribution of X. (Note: the probability of success is 18/35; start by verifying this).
5. Let X be a random variable with the cumulative distribution function (cdf) F = F (x) and let
α, β be real numbers with α ̸= 0. Determine the cdf of the random variable αX + β. (Keep in mind
that F (x) = P (X ≤ x) and you cannot assume that F is continuous.)
6. For p ∈ (0, 1), let x(p) be the smallest number of people so that there is a better than 100 · p%
chance to have at least two born on the same day. Derive an approximate expression for x(p) [it is√
2n ln(1/q), n = 365, q = 1− p] and sketch the graph of the function x = x(p).
Homework 6.
1. Consider the function
C(2x− x2) 0 < x < 2
0 otherwise.
(a) Could f be a distribution function? If so, compute the value of C.
(b) Could f be a probability density function? If so, compute the value of C.
2. (I) A stick is broken into two pieces at random. Compute the probability that the ratio of the
longer part to the shorter is at least a, where a > 1. (The length of the stick does not matter; put it
equal to 1 if you want).
(II) A stick is broken in three pieces at random. Compute the probability p that the pieces are sides
of the triangle (that it, the sum of any two is bigger than the third). Consider the following three
possibilities (a) selecting two iid uniforms on the stick [then p = 1/4, best seeing by thinking of the
stick as the height of an equilateral triangle]; (b) breaking the stick at a uniform point in two pieces,
then taking the longer piece and breaking it again [then p = 2(ln 2 − 0.5)]; (c) breaking the stick at a
uniform point in two pieces, then taking one of the pieces with probability proportional to the length
of the piece and breaking it again [then p = 1/4].
3. Given a normal random variable X with mean 10 and variance 36, compute the following prob-
abilities: (a) P (X > 5); (b) P (4 < X < 16); (c) P (X < 8); (d) P (X < 20); (e) P (X > 16). Either
a table or statistical function on your calculator can be used to produce the numerical answers. For
better results, reduce to standard normal before computing the numerical answer.
4. (a) Compute the standard deviation of a normal random variable X if E(X) = 5 and P (X > 9) =
0.2. (b) Compute the expected value of a normal random variable X if var(X) = 3 and P (X > 5) = 0.2.
(c) Determine the mean and the standard deviation of a normal random variable X in each of the two
Case 1 : P (X > 0) =
and P (X < 2) = 1 + 0.8664 Case 2 : P (X < 0) = and P (X < 2) = 1 + 0.8664 Note. The numbers 0.5763 and 0.8664 come directly from a Z-table, and should lead you to the right Z- value. The Z-value corresponding to 0.5763 is ±0.8; to figure out whether you take positive or negative value, draw a picture. (d) Create your own problem similar to (c). What are the constraints on the numbers you are using? 5. Let X be binomial random variable with parameters n = 100 and p = 0.65. Use normal ap- proximation with continuity correction to approximate the following probabilities: (a) P (X ≥ 50); (b) P (60 ≤ X ≤ 70); (c) P (X < 75). 6. The number of miles a certain car can drive before breaking is a random variable X. The car has been driven for 10000 miles. Compute the probability that the car will drive another 20000 if the distribution of X is (a) exponential, with average value 20000; (b) unform on (0, 40000). 7. Let X be exponential random variable with mean 1. Determine the probability density function 8. Let X be uniform on (0, 1). Determine the probability density function of eX . 9. Let X be a continuous random variable, and assume that the cdf FX of X is a strictly increasing function. Identify the distribution of the random variable U = FX(X). What, if anything, changes without the assumption that FX is strictly increasing? What if X is not continuous? Homework 7. 1. Two fair dice are rolled. Define the following random variables: X, the value of the first die; Y , the sum of the two values; Z, the larger of the two values; V , the smaller of the two values. Determine the joint distribution of (a) Z and Y ; (b) X and Y ; (c) Z and V . 2. The joint probability density function of two random variables X and Y is f(x, y) = c(y2 − x2)e−y, −y ≤ x ≤ y, 0 < y < +∞. Compute (a) the value of c; (b) the marginal densities of X and Y ; (c) expected value of X. 3. Two people decide to meet at a certain location. The arrival time of person A is uniformly distributed between 12:15pm and 12:45pm. The arrival time of person B is uniformly distributed between 12pm and 1pm. The two people arrive independently of each other. (a) Compute the probability that the first to arrive will wait at most five minutes. (b) Compute the probability that person A arrives first. [Draw a picture]. 4. Compute the probability that n points, selected randomly and independently on a circle, will be in the same semi-circle. (Suggestion: Fix a point Pi and denote by Ai the event that all points starting with Pi and going clockwise are in the same semi-circle. Argue that Ai and Aj are mutually exclusive and that the probability of each Ai is 2 5. Two points X,Y are selected at random on the interval [0, 1] so that X is uniform on (0, 1/2), Y is uniform on (1/2, 1) and X,Y are independent. Compute P (Y −X > 1/3).
6. Given the joint density f = f(x, y) of two random variables X,Y , decide whether the random
variables are independent:
(a) f(x) =
xe−(x+y), x > 0, y > 0;
0, otherwise
(b) f(x) =
2, 0 < x < y, 0 < y < 1; 0, otherwise 7. Player A’s bowling score in one game is approximately normal with mean 170 and standard deviation 20; player B’s score in one game is approximately normal with mean 160 and standard deviation 15. Assuming that the scores are independent, compute the probability that, in one game, (a) Player A scores higher than player B; (b) The total score is over 350. Homework 8. 1. The joint probability density function of two random variables X and Y is f(x, y) = c(x2 − y2)e−x, 0 < x < +∞, −x ≤ y ≤ x. Compute the conditional distribution of Y given X = x. 2. The joint probability density function of two random variables X and Y fXY (x, y) = Cy if x2 + y2 ≤ 1, |x| ≤ 1, y ≥ 0, 0 otherwise. Determine the value of C and then compute EX,EY,Var(X),Var(Y ),Cov(X,Y ), Cor(X,Y ), fX(x), fY (y), fX|Y (x|y), fY |X(y|x), P (|X| < 1/2|Y = 1/2). 3. Three cars break down on a road of length L, randomly and independently of one another. Given a d < L/2, compute the probability that the distance between any two of the cars is at least d. (Keep in mind that there are six possible arrangements of the cars). 4. The random variables X,Y have the joint density , x2 + y2 ≤ 1 0, otherwise. In other words, the vector (X,Y ) is uniform in the unit disk. Compute the joint density of and tan−1(Y/X). 5. Let U,Z be independent random variables such that U is uniform on (0, 2π) and Z is exponential with mean 1. Show that X = 2Z cosU and Y = 2Z sinU are independent standard normal random variables. 6. Let X,Y be independent random variables, both uniform on (0, 1). Compute the joint density of the following random variables; (a) X + Y, X/Y ; (b)X, X/Y , (c) X + Y, X/(X + Y ). Then repeat the problem w 程序代写 CS代考 加微信: powcoder QQ: 1823890830 Email: powcoder@163.com