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theory of probability

Questions and Answers of Theory Of Probability

27. Verify Equation (6.6), which gives the joint density of X(1) and X(n).
26. Show that the median of a sample of size 2n + 1 from a uniform distribution on (0, 1) has a beta distribution with parameters (n + 1, n + 1).
25. Establish Equation (6.2) by differentiating Equation (6.4).
24. Show that if n people are distributed at random along a road L miles long, then the probability that no 2 people are less than a distance of D miles apart is, when D ≤ L/(n - 1), [1 -
23. Suppose that F(x) is a cumulative distribution function. Show that (a) F(x)and (b) 1 - [1 - F(x)]n are also cumulative distribution functions when n is a positive integer.HINT: Let X1, . . ., Xn
22. The random variables X and Y are said to have a bivariate normal distribution if their joint density function is given by$$f(x, y) = \frac{1}{2\pi\sigma_x\sigma_y\sqrt{1-ho^2}}$$$$\times
21. A rectangular array of me numbers arranged in n rows, each consisting of m columns, is said to contain a saddlepoint if there is a number that is the minimum of its row and the maximum of its
20. Let W be a gamma random variable with parameters (r, B), and suppose that conditional on W = w, X1, X2, …, X, are independent exponential random variables with rate w. Show that the conditional
19. Suppose that W, the amount of moisture in the air on a given day, is a gamma random variable with parameters (t, B). That is, its density is f(w) = Be-Bw (Bw)/(t), w > 0. Suppose also that given
18. Let U denote a random variable uniformly distributed over (0, 1). Compute the conditional distribution of U given that(a) U> a;(b) U
17. Let X1, X2, X3 be independent and identically distributed continuous random variables. Compute(a) P(X₁ > X2|X₁ > X3);(b) P{X₁ > X2|X1 < X₃);(c) P(X1 > X2|X2 X2|X2
16. Consider an experiment that results in one of three possible outcomes, outcome i occurring with probability p,, i = 1, 2, 3. Suppose that independent replications of this experiment are performed
15. If X and Y are independent binomial random variables with identical parameters n and p, show analytically that the conditional distribution of X, given that X+Y=m, is the hypergeometric
14. Suppose that X and Y are independent geometric random variables w same parameter p.(a) Without any compotations, what do you think is the value of P(XiX+Y)?HINT: Imagine that you continually flip
13. In Example 5c we computed the conditional density of a success prob for a sequence of trials when the first trials resulted in suc Would the conditional density change if we actually specified
12. Show that the jointly continuous (discrete) random variables X₁.....independent if and only if their joint probability density (mass) fu f(x) can be written as(....) -8(x)for nonnegative
11. Let X1, X2, X3, X4, X5, be independent continuous random variables a common distribution function F and density functionf, and set 1-P{XX > X3 X4 X5)(a) Show that I does not depend on F.HINT:
10. The lifetimes of batteries are independent exponential random variable having parameter A. A flashlight needs 2 batteries to work. If one flashlight and a stockpile of n batteries, what is the
9. Let X₁, X be independent exponential random variables ha common parameter A. Determine the distribution of min (X1,
8. Let X and Y be independent continuous random variables with res hazard rate functions Ax(1) and Ay(7), and set W=min(X, Y).(a) Determine the distribution function of W in terms of those of X(b)
7. (a) If X has a gamma distribution with parameters (r, A), what is the di tion of cX, c > 0?(b) Show that 1has a gamma distribution with parameters n, A when n is a positive and x2, is a
6. Show analytically (by induction) that X₁ + + X has a negative binomial distribution when the X, i 1,..., n are independent and identically distributed geometric random variables. Also, give a
5. If X and Y are independent continuous positive random variables, express the density function of (a) Z- X/Y and (b) Z XY in terms of the density functions of X and Y. Evaluate these expressions in
4. Solve Buffon's needle problem when L > D.ANSWER:2L Dsin 6) + 20/, where & is such that cos 6 D/L=
3. Suggest a procedure for using Buffon's needle problem to estimate w. Surpris-ingly enough, this was once a common method of evaluating w.
1,..., n, are independent Poisson random variables with respective parame-ters Ap, i 1,...,n.
2. Suppose that the number of events that occur in a given time period is a Poisson random variable with parameter A. If each event is classified as a type i event with probability p,, i = 1, n. p, =
60. Consider an urn containing a balls, numbered 1,..., n.and suppose that k of them are randomly withdrawn. Let X, equal 1 if ball numbered / is removed and let it be 0 otherwise. Show that
59. In Example 8b, let YA+1-+1-Y. Show that Y..... Ya Ye+1 are exchangeable. Note that Y+1 is the number of balls one must observe to obtain a special, ball if one considers the balls in their
58. If X, Y, and Z are independent random variables having identical density functions f(x) = e^x, 0 < x < 0∞, derive the joint distribution of U =X+Y, VX+Z, WY + Z.
57. If Xj and X₂ are independent exponential random variables each having parameter A, find the joint density function of Y, X1 X2 and Y₂ =Xi
56. Repeat Problem 55 when X and Y are independent exponential random vari-ables, each with parameter A = 1.
54. If X and Y have joint density function$$f(x, y) = \frac{1}{x²y²}$$x ≥ 1, y ≥ 1(a) Compute the joint density function of U = XY, V = X/Y.(b) What are the marginal densities?
53. If U is uniform on (0, 2π) and Z, independent of U, is exponential with λ = 1, show directly (without using the results of Example 7b) that X defined by$$X = \sqrt{2Z}cosU$$$$Y =
52. If X and Y are independent random variables both uniformly distributed on (0, 1), find the joint density function of R = √(X² + Y²), Θ = tan⁻¹Y/X.
51. Let X and Y denote the coordinates of a point uniformly chosen in the disk of radius 1 centered at the origin. That is, their joint density is$$f(x,y) = \frac{1}{\pi}$$x² + y² ≤ 1 Find the
50. Derive the distribution of the range of a sample of size 2 from a distribution having density function f(x) = 2x, 0 < x < 1.
49. If X₁, X₂, X₃, X₄, X₅ are independent and identically distributed exponential random variables with the parameter λ, compute(a) P(min(X₁, ..., X₅) ≤ α);(b) P(max(X₁, ...,
48. Consider a sample of size 5 from a uniform distribution over (0, 1). Compute the probability that the median is in the interval (1/2, 1).
47. If 3 trucks break down at points randomly distributed on a road of length L, find the probability that no 2 of the trucks are within a distance d of each other when d ≤ L/2.
45. If X₁, X₂, X₃ are independent random variables that are uniformly distributed over (a, b), compute the probability that the largest of the three is greater than the sum of the other two.
43. The joint density of X and Y is$$f(x, y) = c(x²- y²)e^{-x}$$0 ≤ x < ∞, -x ≤ y ≤ x Find the conditional distribution of Y, given X = x.
42. The joint density function of X and Y is given by$$f(x,y)=xe^{-x(y+1)}$$x > 0, y > 0.(a) Find the conditional density of X, given Y = y, and that of Y, given X = x.(b) Find the density function
41. The joint probability mass function of X and Y is given by$$p(1, 1) = \frac{1}{8}$$$$p(1, 2) = \frac{1}{4}$$$$p(2, 1) = \frac{1}{8}$$$$p(2, 2) = \frac{1}{4}$$(a) Compute the conditional mass
40. Two dice are rolled. Let X and Y denote, respectively, the largest and smallest values obtained. Compute the conditional mass function of Y given X = i, for i = 1, 2,..., 6. Are X and Y
38. In Problem 5, calculate the conditional probability mass function of Y given that(a) Y₂ = 1;(b) Y₂ = 0.
37. In Problem 3, calculate the conditional probability mass function of Y given that(a) Y₂ = 1;(b) Y₂ = 0.
36. In Problem 4, calculate the conditional probability mass function given that(a) X2 = 1;(b) X₂ = 0.--- OCR End ---
35. In Problem 2, calculate the conditional probability mass function given that(a) X21;(b) X2 0.
34. According to the U.S. National Center for Health Statistics, 25.2 perc males and 23.6 percent of females never eat breakfast. Suppose that ra samples of 200 men and 200 women are chosen.
33. Jill's bowling scores are approximately normally distributed with mea and standard deviation 20, while Jack's scores are approximately non distributed with mean 160 and standard deviation 15. If
32. The gross weekly sales at a certain restaurant is a normal random va with mean $2200 and standard deviation $230. What is the probabilit(a) the total gross sales over the next 2 weeks exceeds
31. The monthly worldwide average number of airplane crashes of comm airlines is 2.2. What is the probability that there will be(a) more than 2 such accidents in the next month;(b) more than 4 such
30. The expected number of typographical errors on a page of a certain mag is .2. What is the probability that an article of 10 pages contains (a)(b) 2 or more typographical errors? Explain your
29. When a current / (measured in amperes) flows through a resistance R sured in ohms), the power generated is given by WIR (measur watts). Suppose that I and R are independent random variables with
28. If X₁ and X2 are independent exponential random variables with respective parameters A, and A₂, find the distribution of Z X1/X2. Also compute P(X₁ < X₂).
27. If X is uniformly distributed over (0, 1) and Y is exponentially distributed with parameter A1, find the distribution of (a) ZX + Y and (b) Z =X/Y. Assume independence.
25. Suppose that 10º people arrive at a service station at times that are independent random variables, each of which is uniformly distributed over (0, 10°). Let N denote the number that arrive in
24. Consider independent trials each of which results in outcome i, i = 0, 1,..., k with probability p Σ 1. Let N denote the number of trials 1-0 needed to obtain an outcome that is not equal to 0,
23. The random variables X and Y have joint density function.$$f(x, y) = 12xy(1-x)$$and equal to 0 otherwise.(a) Are X and Y independent?(b) Find E[X].(e) Find E[Y].(d) Find Var(X).(e) Find Var(Y).0
22. The joint density function of X and Y is$$f(x, y) = \begin{cases}x+y & 0
21. Let$$f(x, y) = 24xy, \qquad 0 \le x \le 1, \ 0 \le y \le 1, \ 0 \le x + y \le 1 $$and let it equal 0 otherwise.(a) Show that *f(x, y)* is a joint probability density function.(b) Find E(X).(c)
20. The joint density of X and Y is given by$$f(x, y) =\begin{cases}xe^{-(x+y)} &\text{x > 0, y > 0} \\0 &\text{otherwise}\end{cases}$$Are X and Y independent? What if f(x, y) were given by$$f(x, y)
19. In Problem 18 find the probability that the 3 line segments from 0 to X, from X to Y, and from Y to L could be made to form the three sides of a triangle. (Note that three line segments can be
18. Two points are selected randomly on a line of length L so as to be on sides of the midpoint of the line. [In other words, the two points X and Y are independent random variables such that X is
17. Three points X1, X2, X3 are selected at random on a line L. What is the probability that X₂ lies between X1 and X3?
16. Suppose that n points are independently chosen at random on the perimeter of a circle, and we want the probability that they all lie in some semicircle.(That is, we want the probability that
15. The random vector (X, Y) is said to be uniformly distributed over a region R in the plane if, for some constantc, its joint density is$$f(x, y) =\begin{cases}c &\text{if } (x, y) \in R \\0
14. An ambulance travels back and forth, at a constant speed, along a road of length L. At a certain moment of time an accident occurs at a point uniformly distributed on the road. [That is, its
12. The number of people that enter a drugstore in a given hour is a Poisson random variable with parameter λ = 10. Compute the conditional probability that at most 3 men entered the drugstore,
11. A television store owner figures that 45 percent of the customers entering his store will purchase an ordinary television set, 15 percent will purchase a color television set, and 40 percent will
10. The joint probability density function of X and Y is given by$$f(x, y) = e^{-(x+y)}, \qquad 0 \le x < \infty, \ 0 \le y < \infty$$Find (a) P(X ≤ Y) and (b) P(X < a).
9. The joint probability density function of X and Y is given by$$f(x, y) = \frac{6}{7} (x^2 + \frac{x}{2}y)$$0 < x < 1, 0 < y < 2(a) Verify that this is indeed a joint density function.(b) Compute
8. The joint probability density function of X and Y is given by$$f(x, y) = c(y^2x^2)e^{-y}$$-y ≤ x ≤ y, 0 < y < ∞(a) Find c.(b) Find the marginal densities of X and Y.(c) Find E[X].
7. Consider a sequence of independent Bernoulli trials, each of which is a success with probability p. Let X1 be the number of failures preceding the first success, and let X2 be the number of
6. A bin of 5 transistors is known to contain 2 that are defective. The transistors are to be tested, one at a time, until the defective ones are identified. Denote by N1 the number of tests made
5. Repeat Problem 3a when the ball selected is replaced in the urn before the next selection.
4. Repeat Problem 2 when the ball selected is replaced in the urn before the next selection.
3. In Problem 2, suppose that the white balls are numbered, and let Yi equal 1 if the ith white ball is selected and 0 otherwise. Find the joint probability mass function of(a) Y1, Y2(b) Y1, Y2, Y3
1. Two fair dice are rolled. Find the joint probability mass function of X and Y when(a) X is the largest value obtained on any die and Y is the sum of the values;(b) X is the value on the first die
16. A standard Cauchy random variable has density function$$f(x) = \frac{1}{\pi(1 + x^2)}$$-∞ < x < ∞If X is a standard Cauchy random variable, show that 1/X is also a star Cauchy random
14. Suppose that the cumulative distribution function of the random variab is given by$$F(x) = 1-e^{-x^2}$$x>0 Evaluate (a) P(X > 2); (b) P(1 < X < 3); (c) the hazard rate funk of F; (d) E[X]; (e)
13. At a certain bank, the amount of time that a customer spends being se by a teller is an exponential random variable with mean 5 minutes. If t is a customer in service when you enter the bank,
12. The following table uses 1992 data concerning the percentages of male and female fulltime workers whose annual salaries fall in different ranges.Barmings range Percentage of females Percentage of
11. The annual rainfall in Cleveland, Ohio is approximately a normal random variable with mean 40.2 inches and standard deviation 8.4 inches. What is the probability that(a) next year's rainfall will
10. The life of a certain type of automobile tire is normally distributed with mean 34,000 miles and standard deviation 4000 miles.(a) What is the probability that such a tire lasts over 40,000
9. Suppose that the travel time from your home to your office is normally distributed with mean 40 minutes and standard deviation 7 minutes. If you want to be 95 percent certain that you will not be
8. A randomly chosen IQ test taker obtains a score that is approximately a normal random variable with mean 100 and standard deviation 15. What is the probability that the test score of such a person
7. To be a winner in the following game, you must be successful in three successive rounds. The game depends on the value of U, a uniform random variable on (0, 1). If U > 1, then you are successful
6. Your company must make a sealed bid for a construction project. If you succeed in winning the contract (by having the lowest bid), then you plan to pay another firm 100 thousand dollars to do the
5. The random variable X is said to be a discrete uniform random variable on the integers 1, 2,..., n if$$P(X=i) = \frac{1}{n}$$ (i = 1, 2,..., n)For any nonnegative real number x, let Int(x)
4. The random variable X has probability density function$$f(x) = \begin{cases}ax + bx^2 & 0 < x < 1 \\0 & otherwise\end{cases}$$If E[X] = 6, find (a) P{X ≤ 1/2} and (b) Var(X).
3. For some constantc, the random variable X has probability density function$$f(x) = \begin{cases}cx^4 & 0 < x < 2 \\0 & otherwise\end{cases}$$Find (a) E[X] and (b) Var(X).
2. For some constantc, the random variable X has probability density function$$f(x) = \begin{cases}cx^2 & 0 < x < 1 \\0 & otherwise\end{cases}$$Find (a) c and (b) P(X > x), 0 < x < 1.
1. The number of minutes of playing time of a certain high school basketball player in a randomly chosen game is a random variable whose probability density function is given below..050 -.025 -10 20
32. Prove Theorem 7.1 when g(x) is a decreasing function.SELF-TEST PROBLEMS AND EXERCISES
31) Let X and Y be independent random variables that are both equally likely to be either 1, 2,..., (10)N, where N is very large. Let D denote the greatest common divisor of X and Y, and let Qk = P(D
29. Let X have probability density fx. Find the probability density function of the random variable Y, defined by Y = ax + b.
28. Let X be a continuous random variable having cumulative distribution function F. Define the random variable Y by Y = F(X). Show that Y is uniformly distributed over (0, 1).
26. If X is uniformly distributed over (a, b), what random variable, having a linear relation with X, is uniformly distributed over (0, 1)?

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