EXERCISES 7. Let X be a random variable with pdf f(x) = 4x if 0...

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EXERCISES 7. Let X be a random variable with pdf f(x) = 4x³ if 0<x< 1 and zero otherwise. Use the cumulative (CDF) technique to determine the pdf of each of the following random variables: (a) Y = X4. (b) W = ex. (c) Z = In X. (d) U = (X-0.5)². 2. Let X be a random variable that is uniformly distributed, X technique to determine the pdf of each of the following: (a) Y = X ¹/4. (b) W = e-X. (c) Z=1-e-X. (d) U = X(1-X). UNIF(0, 1). Use the CDF 3. The measured radius of a circle, R, has pdf f(r) = 6r(1-r), 0 <r < 1. (a) Find the distribution of the circumference. (b) Find the distribution of the area of the circle. EXERCISES 4. 11 X is Weibull distributed, X-WEI(0, p). find both the CDF and pdf of each of the following: (a) Y-(X/0). (b) I-In X. (c) Z = (In X)². 5. Prove Theorem 6.3.4, assuming that the CDF Fix) is a one-to-one function 6. Let X have the pdf given in Exercise 1. Find the transformation y = x) such that Y(X) UNIF(0, 1). 7. Let X UNIF(0, 1). Find transformations y = G₁(u) and w= G(u) such that N (a) Y = G,(U) ~ EXP(1). (b) WG₂(U) BIN(3, 1/2). 8. Rework Exercise 1 using transformation methods. 9. Rework Exercise 2 using transformation methods. 10. Suppose X has pdf f(x) = (1/2) exp(-x) for all -00 < x < 0. (a) Find the pdf of Y = IX). (b) Let W = 0 if X <0 and W = 1 if X > 0. Find the CDF of W. 71. IX-BIN(n, p), then find the pdf of Y = n-X. 12. IX-NB(r, p), then find the pdf of Y = X-r. 13. Let X have pdf f(x)=x2/24; -2 < x <4 and zero otherwise. Find the pdf of Y - X². 14. Let X and Y have joint pdf f(x, y)=4e-2; 0<x< 0,0 < y<co, and zero otherwise. (a) Find the CDF of W X + Y. = (b) Find the joint pdf of U = X/Y and V = X. (c) Find the marginal pdf of U. 227 15. If X, and X, denote a random sample of size 2 from a Poisson distribution, X-POI(AL find the pdf of Y = X₁ + X₂. 16. Let X, and X₂ denote a random sample of size 2 from a distribution with pdf f(x)=1/x²; 1<x<00 and zero otherwise. (a) Find the joint pdf of U=X, X, and V-X₁. (b) Find the marginal pdf of U 17. Suppose that X, and X, denote a random sample of size 2 from a gamma distribution, X → GAMZ, 1/2). (a) Find the pdf of Y = √√X₁ + X₂- (b) Find the pdf of W-X₁/X₂- 18. Let X and Y have joint pdf f(x, y) = ; 0<x<y< and zero otherwise. (a) Find the joint pdf of SX + Y and T - X CHAPTER 6 FUNCTIONS OF RANDOM VARIABLES (b) Find the marginal pdf of T. (c) Find the marginal pdf of S. 19. Suppose that X₁, X,...., X, are independent random variables and let Y₁ = u(X) for i=1,2,..., k. Show that Y, Y., Y, are independent. Consider only the case where X, is continuous and y, = u(x) is one-to-one. Hint: If x= wy) is the inverse transformation, then the Jacobian has the form J- w (v) 20. Prove Theorem 5.4.5 in the case of discrete random variables X and Y. Hint: Use the transformation s= x and t= g(x)y. 21. Suppose X and Y are continuous random variables with joint pdf f(x, y)2(x + y) if 0<x<y<1 and zero otherwise. (a) Find the joint pdf of the variables S = X and T = XY. (b) Find the marginal pdf of T. 22. As in Exercise 2 of Chapter 5 (page 189), assume the weight (in ounces) of a major league baseball is a random variable, and recall that a carton contains 144 baseballs. Assume now that the weights of individual baseballs are independent and normally distributed with mean = 5 and standard deviation = 2/5, and let T' represent the total weight of all baseballs in the carton. Find the probability that the total weight of baseballs in a carton is at most 725 ounces. 23. Suppose that X₁, X₂, X, are independent random variables, and let Y= X₁ + X₂ ++ X₁. If X, GEO(p), then find the MGF of Y. What is the distribution of Y N 24. Let X₁, X₁,..., X₁o be a random sample of size = 10 from an exponential distribution with mean 2, X,~ EXP(2). D 10 (a) Find the MGF of the sum Y= X.. (=1 (b) What is the pdf of Y? 31 25. Let X₁, X₂, X,, and X, be independent random variables. Assume that X₁, X₁, and X₂ each are Poisson distributed with mean 5, and suppose that Y= X₁ + X₂ + X₁ + X₂ ~ POI(25) (a) What is the distribution of X,? (b) What is the distribution of W X₁ + X₂7 W 26. Let X, and X, be independent negative binomial random variables, X, - NB(r,. p) and X₁-NB(r₂. p). (a) Find the MGF of Y - X, + X₂ (b) What is the distribution of Y? 27. Recall that Y-LOGN(, ²) if in Y-N.o¹). Assume that Y, LOGNU.). 1-1,...,n are independent. Find the distribution of: (2) Y (b) Y (c) Y₂/Y₂ (d) Find E [x] 28. Let X, and X, be a random sample of size = 2 from a continuous distribution with pdf of the form f(x) = 2x if 0<x< 1 and zero otherwise. (a) Find the marginal pdfs of the smallest and largest order statistics, Y, and Y₂. (b) Find the joint pdf of Y, and Y₂. (c) Find the pdf of the sample range R = Y₂ - Y₁. 29. Consider a random sample of size n from a distribution with pdf f(x)=1/x² if 1 < x < 00: zero otherwise. (a) Give the joint pdf of the order statistics. (b) Give the pdf of the smallest order statistic, Y₁. (c) Give the pdf of the largest order statistic, Y,. (d) Derive the pdf of the sample range, R = Y, Y₁, for n = 2. (e) Give the pdf of the sample median, Y,, assuming that n is odd so that r = (n + 1)/2. 30. Consider a random sample of size n = 5 from a Pareto distribution, X, ~ PAR(1, 2). (a) Give the joint pdf of the second and fourth order statistics. Y₂ and Y. (b) Give the joint pdf of the first three order statistics, Y₁, Y₂, and Y₂. (c) Give the CDF of the sample median, Y,. 31. Consider a random sample of size n from an exponential distribution, X, EXP(1). Give 21 the pdf of each of the following: (a) The smallest order statistic, Y₁. (b) The largest order statistic, Y. (c) The sample range, R = Y - Y₁. (d) The first order statistics. Y..... Y,. 32. A system is composed of five independent components connected in series. (a) If the pdf of the time to failure of each component is exponential, X,~ EXP(1), then fr give the pdf of the time to failure of the system. (b) Repear (a), but assume that the components are connected in parallel. (c) Suppose that the five-component system fails when at least three components fail. Give the pdf of the time to failure of the system. (d) Suppose that n independent components are not distributed identically, but rather X, EXP(0). Give the pdf of the time to failure of a series system in this case. 33. Consider a random sample of size n from a geometric distribution, X, GEO(p). Give the CDF of each of the following: (a) The minimum, Y₁ (b) The kth smallest, Ya- (c) The maximum, Y,. (d) Find P[Y, <1]. 34. Suppose X, and X₂ are continuous random variables with joint pdf f(x₁, x₂). Prove u(x1, x₂), Y2 = X₂ is one-to-one. Hint: Theorem 5.2.1 assuming the transformation y, = First derive the marginal pdf of Y₁ = u(X₁, X₂) and show that E(Y₁) = fy₁ fr. (1) dy - ff*(*. *₂) f(x u(x₁, x₂) f(x₁, x₂) dx, dx₂. Use a similar proof in the case of discrete random variables. Notice that proofs for the cases of k variables and transformations that are not one-to-one are similar but more complicated. 35. Suppose X₁, X₂ are independent exponentially distributed random variables, X,~ EXP(0), and let Y = X₁ - X₂. (a) Find the MGF of Y. (b) What is the distribution of Y? 36. Show that if X₁..... , X, are independent random variables with FMGFs G₁(t), G and Y = X₁ + ... + X₁, respectively, then the FMGF of Y is Gy(t) = G(0) G.(¹) EXERCISES 7. Let X be a random variable with pdf f(x) = 4x³ if 0<x< 1 and zero otherwise. Use the cumulative (CDF) technique to determine the pdf of each of the following random variables: (a) Y = X4. (b) W = ex. (c) Z = In X. (d) U = (X-0.5)². 2. Let X be a random variable that is uniformly distributed, X technique to determine the pdf of each of the following: (a) Y = X ¹/4. (b) W = e-X. (c) Z=1-e-X. (d) U = X(1-X). UNIF(0, 1). Use the CDF 3. The measured radius of a circle, R, has pdf f(r) = 6r(1-r), 0 <r < 1. (a) Find the distribution of the circumference. (b) Find the distribution of the area of the circle. EXERCISES 4. 11 X is Weibull distributed, X-WEI(0, p). find both the CDF and pdf of each of the following: (a) Y-(X/0). (b) I-In X. (c) Z = (In X)². 5. Prove Theorem 6.3.4, assuming that the CDF Fix) is a one-to-one function 6. Let X have the pdf given in Exercise 1. Find the transformation y = x) such that Y(X) UNIF(0, 1). 7. Let X UNIF(0, 1). Find transformations y = G₁(u) and w= G(u) such that N (a) Y = G,(U) ~ EXP(1). (b) WG₂(U) BIN(3, 1/2). 8. Rework Exercise 1 using transformation methods. 9. Rework Exercise 2 using transformation methods. 10. Suppose X has pdf f(x) = (1/2) exp(-x) for all -00 < x < 0. (a) Find the pdf of Y = IX). (b) Let W = 0 if X <0 and W = 1 if X > 0. Find the CDF of W. 71. IX-BIN(n, p), then find the pdf of Y = n-X. 12. IX-NB(r, p), then find the pdf of Y = X-r. 13. Let X have pdf f(x)=x2/24; -2 < x <4 and zero otherwise. Find the pdf of Y - X². 14. Let X and Y have joint pdf f(x, y)=4e-2; 0<x< 0,0 < y<co, and zero otherwise. (a) Find the CDF of W X + Y. = (b) Find the joint pdf of U = X/Y and V = X. (c) Find the marginal pdf of U. 227 15. If X, and X, denote a random sample of size 2 from a Poisson distribution, X-POI(AL find the pdf of Y = X₁ + X₂. 16. Let X, and X₂ denote a random sample of size 2 from a distribution with pdf f(x)=1/x²; 1<x<00 and zero otherwise. (a) Find the joint pdf of U=X, X, and V-X₁. (b) Find the marginal pdf of U 17. Suppose that X, and X, denote a random sample of size 2 from a gamma distribution, X → GAMZ, 1/2). (a) Find the pdf of Y = √√X₁ + X₂- (b) Find the pdf of W-X₁/X₂- 18. Let X and Y have joint pdf f(x, y) = ; 0<x<y< and zero otherwise. (a) Find the joint pdf of SX + Y and T - X CHAPTER 6 FUNCTIONS OF RANDOM VARIABLES (b) Find the marginal pdf of T. (c) Find the marginal pdf of S. 19. Suppose that X₁, X,...., X, are independent random variables and let Y₁ = u(X) for i=1,2,..., k. Show that Y, Y., Y, are independent. Consider only the case where X, is continuous and y, = u(x) is one-to-one. Hint: If x= wy) is the inverse transformation, then the Jacobian has the form J- w (v) 20. Prove Theorem 5.4.5 in the case of discrete random variables X and Y. Hint: Use the transformation s= x and t= g(x)y. 21. Suppose X and Y are continuous random variables with joint pdf f(x, y)2(x + y) if 0<x<y<1 and zero otherwise. (a) Find the joint pdf of the variables S = X and T = XY. (b) Find the marginal pdf of T. 22. As in Exercise 2 of Chapter 5 (page 189), assume the weight (in ounces) of a major league baseball is a random variable, and recall that a carton contains 144 baseballs. Assume now that the weights of individual baseballs are independent and normally distributed with mean = 5 and standard deviation = 2/5, and let T' represent the total weight of all baseballs in the carton. Find the probability that the total weight of baseballs in a carton is at most 725 ounces. 23. Suppose that X₁, X₂, X, are independent random variables, and let Y= X₁ + X₂ ++ X₁. If X, GEO(p), then find the MGF of Y. What is the distribution of Y N 24. Let X₁, X₁,..., X₁o be a random sample of size = 10 from an exponential distribution with mean 2, X,~ EXP(2). D 10 (a) Find the MGF of the sum Y= X.. (=1 (b) What is the pdf of Y? 31 25. Let X₁, X₂, X,, and X, be independent random variables. Assume that X₁, X₁, and X₂ each are Poisson distributed with mean 5, and suppose that Y= X₁ + X₂ + X₁ + X₂ ~ POI(25) (a) What is the distribution of X,? (b) What is the distribution of W X₁ + X₂7 W 26. Let X, and X, be independent negative binomial random variables, X, - NB(r,. p) and X₁-NB(r₂. p). (a) Find the MGF of Y - X, + X₂ (b) What is the distribution of Y? 27. Recall that Y-LOGN(, ²) if in Y-N.o¹). Assume that Y, LOGNU.). 1-1,...,n are independent. Find the distribution of: (2) Y (b) Y (c) Y₂/Y₂ (d) Find E [x] 28. Let X, and X, be a random sample of size = 2 from a continuous distribution with pdf of the form f(x) = 2x if 0<x< 1 and zero otherwise. (a) Find the marginal pdfs of the smallest and largest order statistics, Y, and Y₂. (b) Find the joint pdf of Y, and Y₂. (c) Find the pdf of the sample range R = Y₂ - Y₁. 29. Consider a random sample of size n from a distribution with pdf f(x)=1/x² if 1 < x < 00: zero otherwise. (a) Give the joint pdf of the order statistics. (b) Give the pdf of the smallest order statistic, Y₁. (c) Give the pdf of the largest order statistic, Y,. (d) Derive the pdf of the sample range, R = Y, Y₁, for n = 2. (e) Give the pdf of the sample median, Y,, assuming that n is odd so that r = (n + 1)/2. 30. Consider a random sample of size n = 5 from a Pareto distribution, X, ~ PAR(1, 2). (a) Give the joint pdf of the second and fourth order statistics. Y₂ and Y. (b) Give the joint pdf of the first three order statistics, Y₁, Y₂, and Y₂. (c) Give the CDF of the sample median, Y,. 31. Consider a random sample of size n from an exponential distribution, X, EXP(1). Give 21 the pdf of each of the following: (a) The smallest order statistic, Y₁. (b) The largest order statistic, Y. (c) The sample range, R = Y - Y₁. (d) The first order statistics. Y..... Y,. 32. A system is composed of five independent components connected in series. (a) If the pdf of the time to failure of each component is exponential, X,~ EXP(1), then fr give the pdf of the time to failure of the system. (b) Repear (a), but assume that the components are connected in parallel. (c) Suppose that the five-component system fails when at least three components fail. Give the pdf of the time to failure of the system. (d) Suppose that n independent components are not distributed identically, but rather X, EXP(0). Give the pdf of the time to failure of a series system in this case. 33. Consider a random sample of size n from a geometric distribution, X, GEO(p). Give the CDF of each of the following: (a) The minimum, Y₁ (b) The kth smallest, Ya- (c) The maximum, Y,. (d) Find P[Y, <1]. 34. Suppose X, and X₂ are continuous random variables with joint pdf f(x₁, x₂). Prove u(x1, x₂), Y2 = X₂ is one-to-one. Hint: Theorem 5.2.1 assuming the transformation y, = First derive the marginal pdf of Y₁ = u(X₁, X₂) and show that E(Y₁) = fy₁ fr. (1) dy - ff*(*. *₂) f(x u(x₁, x₂) f(x₁, x₂) dx, dx₂. Use a similar proof in the case of discrete random variables. Notice that proofs for the cases of k variables and transformations that are not one-to-one are similar but more complicated. 35. Suppose X₁, X₂ are independent exponentially distributed random variables, X,~ EXP(0), and let Y = X₁ - X₂. (a) Find the MGF of Y. (b) What is the distribution of Y? 36. Show that if X₁..... , X, are independent random variables with FMGFs G₁(t), G and Y = X₁ + ... + X₁, respectively, then the FMGF of Y is Gy(t) = G(0) G.(¹)

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