Due Tme: 5:00PM Thursda, on /09/006 Problem (8 ponts) Book problem 45 Let U = X + and V = X, then the jont pmf of ( UV, ) s θ λ θ e λ e f( u, ) = ( = 0, ; u =, +, )! ( u )! Then f( u, ) u θ λ f ( x x+ ) = = ( u x, x) f( ) = + =, θ + λ θ + λ θ whch s Bnomal( x+, θ + λ ) Smlarl, ou can get f( u, ) u λ ϑ f ( x+ ) = = ( u x, ) f( u) = + = θ + λ θ + λ Problem (5 ponts) Book problem 46 (a) The support set of ( UV, ) s {( u, ) : u=,, ; = 0, ±, ±, } For ( u, ) and > 0, f( u, ) P( U u, V ) P( u, X ) P( X u, u) p ( p) u For ( u, ) and = 0, + = = = = = = = = + = = For ( u, ) and < 0, f( u, ) P( X u, u) p ( p) u = = = = Thus f( u, ) P( X u, u ) p ( p) u = = = = u f( u, ) = p ( p) ( p) ( u =, ; = 0, ±, ) Therefore, U and V are ndependent (b) Let Z = X /( X + ) and V = X, then X = V and = V / Z V The jont pmf of ( Z, V ) s + / z / z fzv, ( z, ) = P( Z = z, V = ) = P( X =, = / z ) = p ( p) = p ( p) for =, and z s taken from all rreducble proper fracton Thus the margnal pmf of Z s
/ z / z p ( p) fz ( z) = p ( p) =, where the support set of Z s all rreducble = / z ( p) ( p) proper fracton (c) Let U = X, V = X +, then the jont pmf of ( UV, ) s f ( u, ) = PX ( = u, = u) = p( p) I ( ui ) ( u) u+ u UV, {, } {, } = p ( p) I{, } ( u) I{, } ( u), the support set of ( UV, ) s u =, ; = u+, Problem 3 (0 ponts) Book problem 40 (a) Let A0 = {( x, x) : x = 0}, A = {( x, x) : x > 0}, A = {( x, x) : x < 0}, then PA ( 0) = 0 and on A and A, the transformaton s one to one On : x =, A On : x =, A Therefore, the jont pdf of (, ) s x = ( ), J = x = ( ), J = f (, ) = exp( )(0 < <, < < ), πσ σ (b) and are the functon of dstance and angle The ndependence between them means the dstance and the angle are ndependent ; Problem 4 (0 ponts) Book problem 44 The transformaton s one to one from (0, ) (0, ) to (0, ) (0,) and x = zz, = z( z) and J = z The jont pdf of Z and Z are Thus, Z and Z are ndependent f ( z, z ) = ( z z ) e ( z ( z )) Γ() r Γ() s e z r+ s z r s = z e z ( z) Γ() r Γ() s r zz s z( z) Z, Z
Problem 5 ( ponts) Book problem 46 For z > 0, PZ ( zw, = 0) = PZ ( = mn( X, ) zz, = ) z x/ λ / µ λ (/ λ + / µ ) = P ( z, X) = e e dxd= ( e ) 0 λµ λ + µ Smlarl, we hae In addton, and Thus, Z and W are ndependent µ PZ zw e λ + µ (/ λ + / µ ) (, = ) = ( ) λ PW ( = 0) = P ( X) = = PW ( = ) λ + µ P( Z z) P( Z z, W 0) P( Z z, W ) e (/ λ + / µ ) = = + = = PZ ( zw = 0) = PZ ( zw, = 0) / PW ( = 0) = PZ ( z), PZ ( zw = ) = PZ ( zw, = 0) / PW ( = ) = PZ ( z) Problem 6 (5 ponts) Book problem 43 (a) E ( ) = EE ( ( X)) = EnX ( ) = n/ Var( ) = E( Var( X )) + Var( E( X )) = E( nx ( X ) + Var( nx ) = ne X ne X + n Var X = n n + n = n + n ( ) ( ) ( ) / /3 / /6 / n n (b) f, X(, x) = P( = X = x) fx( x) = x ( x) ( = 0,,, n;0< x< ) n n n Γ ( + ) Γ( n + ) (c) f ( ) = x ( x) dx ( 0,,, n) 0 = = = Γ ( n+ ) n+ Problem 7 (5 ponts) (000 Qualf Exam Problem ) () Let X ~ U (0,) and let = a(ln( X)) Fnd the pdf for the random arable () Let the random arables X f x = e for 0 x < x / λ ~ X ( ) λ () Fnd the moment generatng functon for each random arable 3
() Fnd the dstrbuton functon for = X+ X n x n x (3) Let X ~ fx ( x n, p) = p ( p) ( x= 0,,, n) and let x α β p~ g( p α, β) = p ( p) / B( α, β), 0 p, α > 0, β > 0, where Γ( a) Γ( β ) B( αβ, ) = Fnd EX Γ ( α + β ) () The pdf of s f( ) = exp( ) I( aln(), ) ( ) a a () M X () t = λ t Let = X+ X, Z = X, then z z fz, (, z) = exp( )exp( )(0 < z < ) λλ λ λ Thus, f( ) = f, Z(, z) dz = [exp( ) exp( )] 0 λ λ λ λ For λ = λ, f ( ) = exp( ) λ λ λ, whch s also equal to (3) E( X) = E( E( X P)) = E( np) = n α α + β lm λ λ 0 [exp( ) exp( )] λ λ λ λ Problem 8 (5 ponts) (00 October Qualf Exam Problem 3) () Suppose that X s the concentraton (n parts per mllon) of a certan arborne pollutant, and suppose that the random arable = ln X has a dstrbuton that can be adequatel modeled b the denst functon f e β / α ( ) = ( α), < β <,0 < α < () Fnd an explct expresson for F ( ), the cumulate dstrbuton functon (CDF) assocated wth the denst functon f ( ) If α = and β =, use ths CDF to fnd the exact numercal alue of PX ( > 4 X> ) () For the denst functon f ( ) gen aboe, dere an explct expresson for a generatng 4
functon ψ ( t ) that can be used to generate the absolute central moments = E( E r ) for r a non-negate nteger, and then use ψ ( t ) drectl to fnd Var( ), the arance of () Let U be ndependentl and dentcall dstrbuted as Unform(0,) for =, Let X be defned as X = U+ U Show that the probablt denst functon for the random arable X s x,0 x< the followng functon: f( x) = x, x () β exp( ), < β ; α F ( ) = β exp( ), β α PX ( > 4 X> ) = PX ( > 4) / PX ( > ) = P ( > ln(4)) / P ( > ln()) = ( F (ln(4))) /( F (ln())) = 084 E = β and consder Ψ ( t) = E(exp( E t) = E(exp( β t)) = αt d Var( ) = t 0= α dt α t = () Let = U, then the jont pdf of ( X, ) s f( x, ) = (0< <, < x< + ) From ths, t s eas to get the margnal pdf of X r 5