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probability and stochastic modeling

Fundamentals Of Probability With Stochastic Processes 4th Edition Saeed Ghahramani - Solutions

14. Let X1, X2, . . . , Xn be identically distributed, independent, exponential random variables with parameters λ1, λ2, . . . , λn. Prove that Emin(X1, . . . ,Xn)< min????E(X1), . . . ,E(Xn).
13. Suppose that the lifetimes of radio transistors are independent exponential random variables with mean five years. Arnold buys a radio and decides to replace its transistor upon failure two times: once when the original transistor dies and once when the replacement dies. He stops using the
12. Is the following a joint probability density function? if 0 <
11. A point is selected at random from the cubeWhat is the probability that it is inside the sphere inscribed in the cube? N={(x, y, z): a
10. Inside a circle of radius R, n points are selected at random and independently. Find the probability that the distance of the nearest point to the center is at least r.
9. (a) Show that the following is a joint probability density function.(b) Suppose that f is the joint probability density function of X, Y, and Z. Find fX,Y (x, y) and fY (y). In x if 0
8. In a huge office building, the alarm system has n sensors, the lifetime of each being exponential with mean 1/λ, independently of the other ones. If for the next t units of time the alarm system is not checked, what is the probability that at time t only k sensors are working? Assume that
7. Let the joint distribution function of X, Y, and Z be given bywhere λ1, λ2, λ3 > 0.(a) Are X, Y, and Z independent?(b) Find the joint probability density function of X, Y, and Z.(c) Find P(X F(x, y, z) = (1-e) (1-e) (1-e), x, y, z > 0,
6. Let X, Y, and Z be jointly continuous with the following joint probability density function:Are X, Y, and Z independent? Are they pairwise independent? x-1(1+y+2) if x, y, z > 0 f(x, y, z) = otherwise.
5. From the set of families with two children a family is selected at random. Let X1 = 1 if the first child of the family is a girl; X2 = 1 if the second child of the family is a girl;and X3 = 1 if the family has exactly one boy. For i = 1, 2, 3, let Xi = 0 in other cases.Determine if X1, X2, and
4. Let the joint probability density function of X, Y, and Z be given by(a) Find the marginal joint probability density function of X, Y ; X, Z; and Y, Z.(b) Find E(X). 6e-2-y-z f(x, y, z) = if 0
3. Let p(x, y, z) = (xyz)/162, x = 4, 5, y = 1, 2, 3, and z = 1, 2, be the joint probability mass function of the random variables X, Y, Z.(a) Calculate the joint marginal probability mass functions of X, Y ; Y, Z; and X, Z.(b) Find E(Y Z).
2. A jury of 12 people is randomly selected from a group of eight Afro-American, seven Hispanic, three Native American, and 20 white potential jurors. Let A, H, N, andW be the number of Afro-American, Hispanic, Native American, and white jurors selected, respectively.Calculate the joint
Let X be a random point from the interval (0, 1), Y be a random point from(0,X), and Z be a random point from (X, 1). Findf, the joint probability density function of X, Y, and Z.
(a) Prove that the following is a joint probability density function.(b) Suppose that f is the joint probability density function of random variables X, Y, Z, and T. Find fY,Z,T (y, z, t), fX,T (x, t), and fZ(z). if 0
A system has n components, whose lifetimes are exponential random variables with parameters λ1, λ2, . . . , λn, respectively. Suppose that the lifetimes of the components are independent random variables, and the system fails as soon as one of its components fails. Find the probability density
Let p(x, y, z) = k(x2 + y2 + yz), x = 0, 1, 2; y = 2, 3; z = 3, 4.(a) For what value of k is p(x, y, z) a joint probability mass function?(b) Suppose that, for the value of k found in part (a), p(x, y, z) is the joint probability mass function of random variables X, Y, and Z. Find PY,Z(y, z) and
Dr. Shams has 23 hypertensive patients, of whom five do not use any medicine but try to lower their blood pressures by self-help: dieting, exercise, not smoking, relaxation, and so on. Of the remaining 18 patients, 10 use beta blockers and 8 use diuretics. A random sample of seven of all these
10. Let X and Y be two independently selected random numbers from the interval (0, 1).(a) Find the joint probability density function of U = X + Y and V = X/(X + Y ).(b) Find the marginal probability density function of V .
9. Let the joint probability density function of the continuous random variableX and Y be given byFind the probability density function of the random variable 4e-2(x+y) x > 0, y >0 f(x, y) 0 otherwise.
8. Let X be the amount of time a patient with a non-life threatening condition has to spend in an emergency room before being seen by a doctor. Suppose that the emergency room serves such patients with rate Y, where Y is an exponential random variable with mean 1/2. Also suppose that the
7. A parallel system with two components is a system that functions if and only if at least one of its components functions. Let X and Y, the lifetimes of the components, be independent exponential random variables with parameters λ and μ, respectively. Find the probability density function of |X
6. Let the joint probability density function of the continuous random variables X and Y be given byUsing the convolution theorem, find the probability density function of X + Y . (xy/16 1 <
5. A point is selected randomly from the unit disk D =(x, y) : x2 + y2 ≤ 1.(a) Find the conditional probability density function of X, the x-coordinate of the point selected, given that Y, it’s y-coordinate, is y.(b) Find the conditional expected value and the conditional variance of X given
4. A certain admission test, given to prospective graduate applicants, consists of two sections, verbal reasoning and quantitative reasoning. Let 100X and 100Y be the respective verbal and quantitative scores of a randomly selected applicant with an undergraduate 3.0 GPA. Suppose thatX and Y are
3. The lifetime of a particular type of tablet PC is exponential with mean 8 years. The manufacturer sells each tablet PC for $960.00 and offers a warranty to fully refund the customer’s money if the PC fails within a year of purchase and to refund half of the customer’s money if it fails
2. Letbe the joint probability density function of continuous random variables X and Y . Are X and Y independent?Why or why not? 60xy 2 >0, g>0, 2+y1 f(x,y) = otherwise
1. A point is selected at random from the set S =(x, y) : x2 + y2 ≤ 4.(a) What is the probability that it falls inside the set E =(x, y) : x2 + y2 ≤ 1?(b) What is the probability that it falls inside the set F =(x, y) : |x| + |y| < 2?
22. (The Wallet Paradox) Consider the following “paradox” given by Martin Gardner in his book Aha! Gotcha (W. H. Freeman and Company, New York, 1981).Each of two persons places his wallet on the table. Whoever has the smallest amount of money in his wallet, wins all the money in the other
21. Let the joint probability density function of random variables X and Y be given byShow that E(Y | X = x) is a linear function of x while E(X | Y = y) is not a linear function of y. if y
18. If F is the distribution function of a random variable X, is G(x, y) = F(x) + F(y) a joint distribution function?
15. Let the joint probability density function of X and Y be given by(a) Determine the value of c.(b) Determine if X and Y are independent. Jcx(1-x) if 0 x y 1 f(x,y) = otherwise.
13. Let X and Y be continuous random variables with the joint probability density functionFind E(Xn | Y = y), n ≥ 1. f(x, y) = == if y>0, 0 <
12. For=(x, y) : 0 , a region in the plane, letbe the joint probability density function of the random variables X and Y . Find the marginal probability density functions of X and Y, and P(X + Y > 1/2). (3(x+y) f(x, y) if (x, y) = N otherwise
11. Let the joint distribution function of the lifetimes of two brands of lightbulb be given byFind the probability that one lightbulb lasts more than twice as long as the other. F(x,y) < < (1)(1) otherwise.
8. Prove that the following cannot be the joint distribution function of two randomvariables X and Y . F(x,y) = if x + y 1 if x + y < 1.
7. Let X and Y have the joint probability density function below. Determine if E(XY ) =E(X)E(Y ). f(x,y) = 3 + if 0 <
6. Let the joint probability density function of X and Y be given by(a) Determine the value of c.(b) Find the marginal probability density functions of X and Y . C if 0 < y
1. The joint probability mass function of X and Y is given by the following table.(a) Find P(XY ≤ 6).(b) Find E(X) and E(Y ). X Y 1 2 3 2 0.05 0.25 0.15 4 0.14 0.10 0.17 6 0.10 0.02 0.02
2. The joint probability density function of the continuous random variables X and Y is given byFind h, the probability density function of X + Y . 2e-(x+2y) x0, y 0 f(x,y) = otherwise.
1. Let X and Y be two independently selected random numbers from the interval (0, 1).Find the joint probability density function of U = X + Y and V = X − Y .
13. Let X and Y be independent (strictly positive) exponential random variables each with parameter λ. Are the random variables X + Y and X/Y independent?
12. Let X and Y be independent gamma random variables with parameters (n/2, 1/2) and(m/2, 1/2), respectively, where n and m are positive integers. The random variable U = (X/n)(Y/m) is said to have F-distribution with n and m degrees of freedom, and has significant applications in statistics. Find
11. LetX and Y be independent (strictly positive) gamma randomvariableswith parameters(r1, λ) and (r2, λ), respectively. Define U = X + Y and V = X/(X + Y ).(a) Find the joint probability density function of U and V .(b) Prove that U and V are independent.(c) Show that U is gamma and V is beta.
10. Prove that if X and Y are independent standard normal random variables, then X + Y and X − Y are independent random variables. This is a special case of the following important theorem.Let X and Y be independent random variables with a common distribution F. The random variables X + Y and X
9. Let X and Y be independent random variables with common probability density functionFind the joint probability density function of U = X + Y and V = eX. f(x) = if x > 0 elsewhere.
8. Let X and Y be independent random variables with common probability density functionCalculate the joint probability density function of U = X/Y and V = XY. 2 f(x) = if x > 1 elsewhere.
7. All international passengers arriving at a U.S. airport must go through the first stage of an immigration process, which takes an exponential length of time with parameterλ. However, experience shows that for some p, 0 < p < 1, 100p% of the passengers have discrepancies in their travel related
6. Let −1/9 (a) Show that the probability mass function of X + Y is the convolution function of the probability mass functions of X and Y for all c.(b) Show that X and Y are independent if and only if c = 0. Y T -1 0 1 -1 1/9 0 1/9+c 1/9 c 1/9+c 1/9 1/9-c 11/9-c 1/9+c 1/9
5. From the interval (0, 1), two random numbers X and Y are selected independently.Show that the probability density function of their sum is given byBecause of the shape of the probability density function of X + Y, the random variable X + Y is said to have a triangular distribution. (See Figure
4. Let X and Y be continuous random variables with the joint probability density function given byFind the probability density function of U = XY . 8xy 0 < y
3. Let X ∼ N(0, 1) and Y ∼ N(0, 1) be independent random variables. Find the joint probability density function of R = √X2 + Y 2 and = arctan(Y/X). Show that R and are independent. Note that (R,) is the polar coordinate representation of(X, Y ).
Let X and Y be independent exponential random variables, each with parameter λ. Find the distribution function of X + Y .
Let X and Y be two independent uniform random variables over (0, 1);show that the random variables U = cos(2πX)√−2 ln Y and V = sin(2πX)√−2 ln Y are independent standard normal random variables.
Let X and Y be positive independent random variables with the identical probability density function e−x for x > 0. Find the joint probability density function of U = X + Y and V = X/Y .
2. A bivariate probability density function is called copula if its marginal probability density functions are uniform random variables over the interval (0, 1). For a constant αbetween −1 and 1, let X and Y be continuous random variables with the joint probability density function given by f(x,
1. There are 16 pea seeds in a packet available to a gardener. Seven of the peas areWando, 4 are Maestro, and 5 are Lincoln. To test the soil of a specific garden for growing peas, the gardener picks 8 of these seeds at random to plant in that garden. Let X be the number of Lincoln seeds and Y be
24. The lifetimes of batteries manufactured by a certain company are identically distributed with distribution and probability density functions F andf, respectively. Suppose that a battery manufactured by this company is installed at time 0 and begins to operate. If at time s an inspector finds
23. Let X and Y be discrete random variables with joint probability mass function x = 0, 1, 2,..., y = 0, 1, 2,...,x, p(x, y) = 1 ey! (x-y)!" p(x, y) = 0, elsewhere. Find E(Y | X = 2). ==
22. A point (X, Y ) is selected randomly from the triangle with vertices (0, 0), (0, 1), and(1, 0).(a) Find the joint probability density function of X and Y .(b) Calculate fX|Y (x|y).(c) Evaluate E(X | Y = y).
21. Let X and Y be continuous random variables with joint probability density functionFind the conditional expectation of Y given that X = x. f(x, y) = Jn(n-1)(y-x)-2 if 0x y 1 = otherwise.
20. A box contains 10 red and 12 blue chips. Suppose that 18 chips are drawn, one by one, at random and with replacement. If it is known that 10 of them are blue, show that the expected number of blue chips in the first nine draws is five.
19. Cards are drawn from an ordinary deck of 52, one at a time, randomly and with replacement.Let X and Y denote the number of draws until the first ace and the first king are drawn, respectively. Find E(X | Y = 5).
18. LetN(t) : t ≥ 0be a Poisson process. For s < t show that the conditional distribution of N(s) given N(t) = n is binomial with parameters n and p = s/t. Also find the conditional distribution of N(t) given N(s) = k.
17. A point is selected at random and uniformly from the region R =(x, y) : |x| + |y| ≤ 1.Find the conditional probability density function of X given Y = y.
16. In a sequence of independent Bernoulli trials, let X be the number of successes in the first m trials and Y be the number of successes in the first n trials, m < n. Show that the conditional distribution of X, given Y = y, is hypergeometric. Also, find the conditional distribution of Y given X
15. Show that ifN(t) : t ≥ 0is a Poisson process, the conditional distribution of the first arrival time given N(t) = 1 is uniform on (0, t).
14. Leon leaves his office every day at a random time between 4:30 P.M. and 5:00 P.M. If he leaves t minutes past 4:30, the time it will take him to reach home is a random number between 20 and 20 + (2t)/3 minutes. Let Y be the number of minutes past 4:30 that Leon leaves his office tomorrow and X
13. The joint probability density function of X and Y is given by(a) Determine the constant c.(b) Find fX|Y (x|y) and fY |X(y|x).(c) Calculate E(Y | X = x) and Var(Y | X = x). f(x,y): Sce- 0 if x 0, y
12. An actuary working for an insurance company has calculated that the time that it will take for an insured driver to report an accident he or she has been involved in, in days, is a continuous random variable X, with the probability density functionFurthermore, the actuary has discovered that if
11. Let (X, Y ) be a random point from a unit disk centered at the origin. Find P(0 ≤ X ≤ 4/11 | Y = 4/5).
10. First a point Y is selected at random from the interval (0, 1). Then another point X is selected at random from the interval (Y, 1). Find the probability density function of X.
9. A box contains 5 blue, 10 green, and 5 red chips.We draw 4 chips at random and without replacement. If exactly one of them is blue, what is the probability mass function of the number of green balls drawn?
8. Let X and Y be continuous random variables with joint probability density function given byCalculate E(X | Y = y). -2(y+1) f(x,y) = if 0, 0
7. Let X and Y be continuous random variables with joint probability density functionCalculate fX|Y (x|y). x+y if 01, 0 y 1 f(x, y): 0 elsewhere.
6. Let X and Y be independent discrete random variables. Prove that for all y, E(X | Y = y) = E(X). Do the same for continuous random variables X and Y .
5. Let the conditional probability density function of X given that Y = y be given byFind P(1/4 fxy (xy) == 3(x + y) 3y+1 0
4. An unbiased coin is flipped until the sixth head is obtained. If the third head occurs on the fifth flip, what is the probability mass function of the number of flips?
3. Let the joint probability density function of continuous random variables X and Y be given byFind fX|Y (x|y). f(x, y) = if 0
2. Regions A and B are prone to dust storms. An actuary has calculated that the number of dust storms that strike region A, in a year, is binomial with parameters 4 and 0.7.Furthermore, she has observed that if, in a year, n dust storms strike region A, then the number of dust storms striking
Let the joint probability mass function of discrete random variables X and Y be given byFind pX|Y (x|y), P(X = 2 | Y = 1), and E(X | Y = 1). =(x + y). if x 1,2, y = 0,1,2 p(x, y) = 25 otherwise.
The lifetimes of batteries manufactured by a certain company are identically distributed with distribution and probability density functions F andf, respectively. In terms of F,f, and s, find the expected value of the lifetime of an s-hour-old battery.
Let X and Y be continuous random variables with joint probability density functionFind E(X | Y = 2). e-y f(x, y) if y> 0, 0 < x < 1 0 elsewhere.
Let the conditional probability density function of X, given that Y = y, beFind P(X fxy (xy) x+y == 0
First, a point Y is selected at random from the interval (0, 1). Then another point X is chosen at random from the interval (0, Y ). Find the probability density function of X.
Let X and Y be continuous random variables with joint probability density functionFind fX|Y (x|y). 2 (x + y) f(x, y): if 0 <
Suppose that there are two identical closed boxes, one containing twice as much money as the other. David is asked to choose one of these boxes for himself. He picks a box at random and opens it. If, after observing the content of the box, he has the option to exchange it for the other box, should
While rolling a balanced die successively, the first 6 occurred on the third roll. What is the expected number of rolls until the first 1?
Calculate the expected number of aces in a randomly selected poker hand that is found to have exactly two jacks.
Let N(t) be the number of males who enter a certain post office at or prior to time t. Let M(t) be the number of females who enter a certain post office at or prior to t.Suppose that{N(t) : t ≥ 0}and{M(t) : t ≥ 0}are independent Poisson processes with rates λ and μ, respectively. So for all
Let the joint probability mass function of X and Y be given byFind pX|Y (x|y) and P(X = 0 | Y = 2) (x+y) if x 0,1,2, y = 1,2 p(x,y) = otherwise.
2. Let X and Y be independent exponential random variables with parameters λ and μ, respectively.What is the probability that X < Y ?
1. The joint probability density function of the continuous random variables X and Y is given byAre X and Y independent?Why or why not? f(x, y) 0 -2x x0, 0 y 2 otherwise.
30. Let f(x, y) be the joint probability density function of two continuous randomvariables;f is called circularly symmetrical if it is a function of √x2 + y2, the distance of (x, y)from the origin; that is, if there exists a function ϕ so that f(x, y) = ϕ(√x2 + y2 ).Prove that if X and Y are
29. Suppose that X and Y are independent, identically distributed exponential random variables with mean 1/λ. Prove that X/(X + Y ) is uniform over (0, 1).
28. Let X and Y be two independent random points from the interval (0, 1). Calculate the distribution function and the probability density function of max(X, Y )/min(X, Y ).
27. Let the joint probability density function of two random variables X and Y satisfywhere g and h are two functions from R to R. Show that X and Y are independent. f(x,y) = g(x)h(y), -xx
26. Let B and C be two independent randomvariables both having the following probability density function:What is the probability that the quadratic equation X2 + BX + C = 0 has two real roots? 3x2 if 1 < x
25. Let E be an event; the random variable IE, defined as follows, is called the indicator of E:Show that A and B are independent events if and only if IA and IB are independent random variables. IE= if E occurs otherwise.
24. Andy buys an MP3 player and six new AAA batteries. The player is operated with two AAA batteries, and each time a battery dies, Andy replaces that battery with a new one.If the lifetimes of the batteries are independent exponential random variables each with parameter λ, find the expected

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