27 Partial Solutions for h4/24s: Sampling Distributions ( Let X and X 2 be two independent random variables, each with the same probability distribution given as follows. f(x 2 e x/2, x (a Compute the probability distribution function of the new random variable Y X + X 2. (b What type of probability distribution is your answer in (a? S(a φ (t φ 2 (t 2t, for t < 2. E [ e t(x +X 2 ] e t(x +x 2 f(x f(x 2 dx dx 2 ( ( e tx f(x dx e tx 2 f(x 2 dx 2 ( 2t 2 Thus, f Y (y 4 ye y/2, <y< S(b Y X + X 2 has a gamma distribution with parameters α 2andθ 2since M Y (t M X (t M X2 (t. 2t 2t ( 2t 2 (2 Let the moment-generating function of X be φ(t e 3t+2t2, <t< (a Find the mean E(X and variance Var(X. (b Name the distribution of X. (c Write down the p.d.f f(x forx. S2(a E(X 3andVar(X 4. S2(b X N(3, 4, that is, X has a normal distribution with mean 3 and variance 4. S2(c The p.d.f of X is f(x 8π e (x 32 /8, <x<. (3 Let X b(n,pandx 2 b(n 2,p be independent r.v.s. Define Y X + X 2.
28 (a What is M Y (t? (b How is Y distributed? S3(a M Y (t M X (tm X2 (t ( p + pe t n ( p + pe t n 2 ( p + pe t n +n 2 S3(b Y b(n + n 2,p. (4 If X N(μ, σ 2, show that Y (ax + b N(aμ + b, a 2 σ 2, a. S4 Without loss of generality (W.L.O.G., let a>, then P (Y y P (ax + b y P (X y b a (y b/a /(2σ 2 2πσ 2 e (x μ2 dx then f(y F (y 2π(aσ e [y (aμ+b]2 /[2(aσ 2], <y< (5 The joint probability density function for two continuous random variables X and Y is given as follows. βxy <x<2 and <y<4 f(x, y elsewhere (a Determine the constant β to make it a joint p.d.f. (b Determine the conditional probability distribution function f(x y. (c Compute the covariance between X and Y. S5(a β 6, f X(y 4 6 xydx x 2, for <x<2, f Y (y 2 6 xydx y 8, for <y<4. S5(b f(x y f(x,y f(y x xy, f(x, y x y f 2 6 2 8 X(xf Y (y. S5(c Cov(X, Y E[(X μ X (Y μ Y ], where and μ x μ y 2 4 x x 2 dx 4 3 y y 8 dy 8 3,
29 Thus Cov(X, Y E[XY ] μ X μ Y Note that X and Y are uncorrelated. 2 4 6 x2 y 2 dydx 32 9 8 64 44 32 9. (6 Show that the sum of n independent Poisson random variables with respective means λ,λ 2,...,λ n is Poisson with mean λ n i λ i. n S6 Let Y X i,then i E[e ty ] E[e t n ix i ] x,x 2,,x n x,x 2,,x n Π n i [etx i f(x i ] Π n i ( [ e n i (tx i ] f(x f(x 2 f(x n x i e tx i f(x i Π n i eλ i(e t e [ n i λ i](e t Thus, Y has a Poisson distribution with mean λ n i λ i. (7 Let Z i N(,, for i 7 and define W 7 i Zi 2. Find P(.69< W <4.7. S7 Y χ 2 (7, then P (.69 <W <4.7 P (W < 4.7 P (W.69.95.25.925. (8 Assume there are obervations, denoted by x i, i, and each is drawn from a population with a continuous distribution in [,2]. (a Give the formula for the sample mean X and sample variance S 2 in this problem. (b Compute the sample mean and sample variance for X. (c Try to give an approximate distribution for X as best as you can and explain your reason. (d Compute the probability that the sample mean value is larger than.2. Note that you can express your solution with the cumulative standard normal distribution function r Φ(r e x2 /2 dx 2π S8(a X i X i, S 2 99 i (X i X 2.
3 S8(b E(X i 2 (x dx, Var(X 2 ie(xi 2 [E(X i] 2 2 (x2 dx, 2 3 then E(X i E(X i Var(X i Var(X i 3 S8(c By C.L.T. Then X E(X Var(X N(, in probability. P (X >.2 P ( 3(X > 3(.2 Φ(.2 3 Φ(.3464.3645 (9 Let random( be a pseudo random number generator which can randomly generate a real in [,. Let X N(,, and Y N(3, 6. Give algorithms to sample (simulate the distributions of X and Y, respectively. S9 Let R i U(, and R n i R i, by C.L.T., we have R E(R nv ar(r N(, in probability or R/n E(R Var(R /n N(, in probability In application, if we choose n 2, then 2 ( R i 6 X N(, i ( Let X,X 2,...,X 3 be a random sample of size 3 from a Poisson distribution with a mean 2/3. Approximate (a P (5 < 3 i X i 22. (b P (2 3 i X i < 27.
3 S(a P (5 < 3 i X i 22 P (5.5 3 i X i 22.5 P ( (5.5 2 3 3 (2/3 i X i 2 3 (2/3 (22.5 2 3 (2/3 P ( 4.5 2 3 i X i 2 2 2.5 2 Φ( 2.5 2 Φ( 4.5 2 Φ(.559 Φ(.62.788.63.5458 S(b P (2 3 i X i < 27 P ( (2.5 2 3 (2/3 3 i X i 2 3 (2/3 (26.5 2 3 (2/3 P (.5 2 3 i X i 2 2 6.5 2 Φ( 6.5 2 Φ(.5 2 Φ(.4534 Φ(.8.9259.5435.3824 ( Let X,X 2,,X n be a random sample of an exponential distribution with mean θ, that is, f(x θ e x/θ, x >. (a Show that P (min(x,x 2,,X n > 2 e 2n/θ. (b Show that P (max(x,x 2,,X n > 2 ( e 2/θ n. (c Find a minimum n such that P (min(x,x 2,,X n > 2 3% when θ 2. (d Find a minimum n such that P (max(x,x 2,,X n > 2 9% when θ 2. S(a [ ] P [min(x,x 2,,X n > 2] Π n ip (X i > 2 Π n i 2 θ e x/θ dx Π n i [ e 2/θ ] e 2n θ
32 S(b P [max(x,x 2,,X n > 2] P (at least one X i > 2 Π n i P (X i 2 ( 2 Π n i θ e x/θ dx [ e 2/θ ] n S(c Find a minimum n such that P (min(x,x 2,,X n > 2 e 2n/θ e 2n/2.3, that is, e n.3, or n ln(.3 3.566 4 S(d Find a minimum n such that P (max(x,x 2,,X n > 2 ( e 2/θ n [ e 2/2 ] n.9, that is, [ e ] n.9 or[ e ] n.. In other words, we want to find n such that (e +(. /n, or (. /n ( e, or n ln(. ln( e. Thus, n 6. e (2 Let W <W 2 < <W n be the order statistics of a random sample of zise n from the uniform distribution U(,. (a Find the probability density function of W. (b Find the probability density function of W n. (c Use the result of (a to find E(W. (d Use the result of (b to find E(W n. Solution: G r (y P (W r y n kr ( n k [F (y] k [ F (y] n k n kr ( n k [F (y] k [ F (y] n k +[F (y] n Thus the p.d.f. of W r could be derived as g r (y G r (y n! (r!(n r! [F (y]r [ F (y] n r f(y, a < y < b ( In particular, g (y n[ F (y] n f(y, a < y < b g n (y n[f (y] n f(y, a < y < b
33 ( If X i has a U(, distribution, then E(W r r. In particular, g n+ (y n( y n, <y<, E(W ; g n+ n(y ny n, <y<, E(W n n. n+ (2 If X j has an exponential distribution with mean 2, then g (y ne ny, y > and E(W n. *(3 One of the most popular distributions used to model the lifetimes of electric components is the Weibull distribution, whose probability density function is given by f(t αt α e tα, for t >, α >. Determine for which values of α the hazard function of a Weibull random variable is increasing, for which values it is decreasing, and for which values it is constant. *S3 f (t α(α t α 2 e tα α 2 t 2α 2 e tα,thus, f (t > if t < ( /α α α < if t > ( /α α α