EXERCISES 7 Let X be a random variable with pdf f(x) 4x if 0 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) 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 G(u) and w G(u) such that N (a) Y G,(U) EXP(1) (b) WG(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 X 14 Let X and Y have joint pdf f(x, y) 4e 2 0 CHAPTER 6 FUNCTIONS OF RANDOM VARIABLES (b) Find the marginal pdf of T (c) Find the marginal pdf of S 19 Suppose that X, X, , X, are independent random variables and let Y 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 (b) Y (c) Y Y (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 (c) The maximum, Y, (d) Find P Y, 1 34 Suppose X, and X are continuous random variables with joint pdf f(x, x) Prove u(x1, x), Y2 X is one to one Hint Theorem 5 2 1 assuming the transformation y, First derive the marginal pdf of Y u(X, X) and show that E(Y) fy fr (1) dy ff ( ) f(x u(x, x) f(x, x) dx, dx 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 X, X are independent exponentially distributed random variables, X, EXP(0), and let Y X X (a) Find the MGF of Y (b) What is the distribution of Y 36 Show that if X , X, are independent random variables with FMGFs G(t), G and Y X X, respectively, then the FMGF of Y is Gy(t) G(0) G ()

Question: EXERCISES 7. Let X be a random variable with pdf f(x) = 4x if 0 EXERCISES 4. 11 X is Weibull distributed, X-WEI(0, p).

EXERCISES 7. Let X be a random variable with pdf f(x) = 4x if 0 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). 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 = G(u) and w= G(u) such that N (a) Y = G,(U) ~ EXP(1). (b) WG(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. 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

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