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text-76-9 - 76 TRANSFORMATIONS AND EXPECTATIONS Section 2.5...

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Unformatted text preview: 76 TRANSFORMATIONS AND EXPECTATIONS Section 2.5 2.5 Exercises 2.1 In each of the following find the pdf of Y. Show that the pdf integrates to 1. (a) Y:X3 and fx(x) =42m5(1—$}, 0<as< 1 '(b) Y=4X+3andfx(m)=7e_7m,0<cc<oo (c) YzX2 and fX{$) :3O$2(1v~$)2, 0<5§< 1 (See Example A.0.2 in Appendix A.) 2.2 In each of the f0110wing find the pdf of Y. (a) YEX2 andjk(w):1,0<sc<1 (b) Y = —logX and X has pdf {n+m+1)1 at”(1 ~— $)m, O < :r: < 1, mm, positive integers n! ml fXW) = (c) Y 2 ex and X has pdf 1 M a 2 i _ fX(:L‘) = 33 are (“3/ l /2, O < :r < oo, 02 a pOSIthe constant 2.3 Suppose X has the geometric prnf fx(5c) : é a: : O,1,2,.... Determine the probability distribution of Y : X / (X + 1). Note that here both X and Y are discrete random variables. To specify the probability distribution of Y, specify its pmf. 2.4 Let A be a fixed positive constant, and define the function f by f : gAeTA‘” if m 2 0 and f($) = gAeM if CC < O. (a) Verify that f is a pdf. (b) If X is a random variable with pdf given by fix), find P{X < t) for all «’3. Evaluate all integrals. (c) Find P([X[ < t) for all t. Evaluate all integrals. 2.5 Use Theorem 2.1.8 to find the pdf of Y in Example 2.1.2. Show that the same answer is obtained by differentiating the cdf given in (2.1.6). 2.6 In each of the following find the pdf of Y and show that the pdf integrates to 1. (a) fxffcl = % ' (b) fX($) = §(sc+ 1)2, —1 <7$ < 1', Y = 1 —X2 (c) fxfl’) = §{x+1)2,—1<m< 1; Yzl—X2 ifXgOandel—XifX>D e"37F —oo < a: < 00; Y 2..|X|3 7 2.7 Let X have pdf fX(93) i %(zc+ I), —1 S :5 g 2. {a} Find the pdf of Y = X 2. Note that Theorem 2.1.8 is not directly applicable in this problem. (b) Show that- Theorem 2.1.8 remains valid if the sets 240,141, . . . , Ag, contain 26, and apply the extension to solve part (a) using A0 = {3, A1 3 (—1,1), and A2 3 {1, 2). 2.8 In each of the following show that the given function is a cdf and find Fgl U if23<0 (a) FX($)_{1—e_93 ifoO Section 2 finc uni 2.10 In T Cdf disr vari (a) (b) 2.11 Let (a) (b) 2.12 A re angl picti the ( Section 2.5 2.9 7 2.10 2.11 2.12 EXERCISES 77 an am<0 (b) axe): 1/2 if0§r<1 1 — (abs/2) if 1 g :3 _ em/tl if :1: < 0 “)kal_{1—(anm) saga Note that, in part (c), FX is discontinuous but (2.1.13) is still the appropriate definition of F}; 1(y). If the random variable X has pdf are) = {521 find a monotone function such that the random variable Y uniform(0, 1) distribution. I In Theorem 2.1.10 the probability integral transform was proved, relating the uniform cdf to any continuous cdf. In this exercise we investigate the relationship between discrete random variables and uniform random variabies. Let X be a discrete random variable with cdf FX and define the random variable Y as Y = FX (X (a) Prove that Y is stochastically greater than a uniform(0, 1); that is, if U N uniform (0, 1), then 1 < at < 3 otherwise, .. u(X) has-a P(Y>y)2P(U>y):1—y, P(Y>y)>P(U>y)=l-y. forally, 0<y<1, forsomey, 0<y<1. (Recall that stochastically greater was defined in Exercise 1.49.) (b) Equivalently, Show that the cdf of Y satisfies Fy(y) g y for all 0 < y < 1 and Fy(y) < y for some O'< y < 1. (Hint: Let 21:0 be a jump point of FX, and define yo = FX (:30). Show that P(Y 3 yo) = go. Now establish the inequality by considering y = yo + 5. Pictures of the cdfs will help.) Let X have the standard normal pdf, fX = (l/x/2rr)e_32/2_ (a) Find EX2 directly, and then by using the pdf of Y = X 2 from Example 2.1.7 and calculating E Y. (b) Find the pdf of Y = |X], and find its mean and variance. A random right triangle can be constructed in the following manner. Let X be a random angle whose distribution is uniform on (0,7r/2). For each X, construct a triangle as pictured below. Here, Y = height of the randm triangle. For a fixed constant d, find the distribution of Y and E Y. (0’. y) 78 2.13 2.14 2.15 2.16 2.17 2.18 2.19 2.20 2.21 TRANSFORMATIONS AND EXPECTATIONS Section 2.5 Consider a sequence of independent coin flips, each of which has probability p of being heads. Define a random variable X as the length of the run (of either heads or tails) started by the first triai. (For example, X = 3 if either TTTH or HHHT is observed.) Find the distribution of X, and find EX. (a) Let X be a continuous, nonnegative random variabie [f(.r) = O for a: < 0]. Show that EX = foo {1 —FX($)]d:c, where FX(:1_3) is the cdf of X. (b) Let X be a discrete random variable whose range is the nonnegative integers. Show that EX. = i (1 — FX(k.-)), k=0 where FXUc) = P(X g Compare this with part (a). Betteley (1977) provides an interesting addition law for expectations. Let X and Y be any two random variables and define X /\ Y = min(X, Y) and X \/ Y = maX(X,Y). Analogous to the probability law P(A U B) : P(A) + P(B) -— P{A F“: B), show that E(XVY) :EX+EY7E(X/\Y). (Hint: Establish that X + Y = (X V Y) + (X /\ Use the result of Exercise 2.14 to find the mean duration of certain telephone calls, where we assume that the duration, T, of a particular call can be described probabilisu tically by P(T > t) : are—At + (1 —— aye—“t, where a, A, and ,a are constants, 0 < a < 1, A > 0, ,u > O. 7 . A median of a distribution is a value in such that P(X g m) Z % and P (X 2 m) > (If X is continuous, m satisfies 1:; f 0133 = f d9: = Find the median of the following distributions. (a)f(x)=332, 0<m<1 —OO<33<OO (b) no 2 Show that if X is a continuous random variable, then minElX—n] 2 EIX —m|, ,where m is the median of X (see Exercise 2.17). Prove that ' %E(X—n)2=0¢$>EX:a by differentiating the integral. Verify, using calculus, that a : EX is indeed a mini— mum. List the assumptions about FX and fx that are needed. A couple decides to continue to have children until a daughter is born. What is the expected number of children of this couple? (Hint: See Example 1.5.4.) Prove the “two—way” rule for expectations, equation (2.2.5), which says E 9(X) 2 BY, where Y = 9(X). Assume that 9(a) is a monotone function. Section 2 2.22 LE (d) 2.27 LeI Th. the inte Section 2.5 EXERCISES 79 2.22 Let X have the pdf 4 m2e_$2/fi2, 0-<w<oo, fi>0. z iris/Fr (a) Verify that f is a pdf. (b) Find EX and Var X. 2.23 Let X have the pdf fix): (1+:c), e1<$<1. l 2 (a) Find the pdf of Y 2 X2. (b) Find EY and Var Y. 2.24 Compute E X and Var X for each of the following probability distributions. (a) fx(:c)_=a:c“*1,0<m<1,a>07 (b) fx($)=%,x=1,2,...,n,n>0an integer (c) fx(m) mgw —— 1)2, 0 < 2: < 2 2.25 Suppose the pdf )5: of a random variable X is an even function. (fX is an even function if 1”); = f X(-—33) for every Show that ' (a) X and ~X are identically distributed. (b) ill/IXOE) is symmetric about 0. 2.26 Let f(CE) be a pdf and let a be a number such that, for ail e > 0, flat + e) 2 flat — 5). Such a pdf is said to be Symmetric about the point a. (3.) Give three examples of symmetric pdfs. (b) Show that if X m fit), symmetric, then the median of X (see Exercise 2.17) is the number a. (c) Show that if X m f(.’1’.'), symmetric, and EX exists, then EX = a. (d) Show that f = 6"", 33 2 O, is not a symmetric pdf. (e) Show that for the pdf in part (d), the median is less than the mean. 12.27 Let flat) be a pdf, and let a be a number such that if a, 2 m 2 y, then f(a) 2 f(a:) 2 fly), and if a, S :1: g y, then flu) Z Z Such a pdf is called nnimodal with a mode equal to a. (a) Give an example of a unimodal pdf for Which the mode is unique. (b) Give an example of a unimodal pdf for which the mode is not unique. (0) Show that if f is both symmetric (see Exercise 2.26) and unimodal, then the point of symmetry is a mode. (d)_yConsider the pdf f (.13) = eMw, 3: 2 0. Show that this pdf is unimodal. What is its mode? 2.28 Let it", denote the nth central moment of a random variable X. Two quantities of interest, in addition to the mean and variance, are 2 i 2 #3 (Palm rl‘he value as is called the skewness and a4 is called the kurtosis. The skewness measures the lack of symmetry in the pdf (see Exercise 2.26). The kurtosis, although harder to interpret, measures the peakedness or flatness of the pdf. E and a4: 2. #2 Ots ...
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