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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..X3 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) hasa P(Y>y)2P(U>y):1—y,
P(Y>y)>P(U>y)=ly. 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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