STA257- Test #1 Time=2hrs Note: No aids. All questions are of ...

STA257- Test #1Time=2hrsNote: No aids. All questions are of equal value. Please sign this sheetwith your student # and hand it in with your test booklets.1.(a) Let A and B be independent events. Show A c and B c areindependent.(b) Show P(A∪B) ≤ P(A) + P(B) .(c) Let A 1, A 2, ... be events with P(A i) = 1 , i=1,2, ... ShownP( ∩ A i) = 1 , n ≥ 2i=12.(a) Let X be a rv and c>0 a constant. Show P(|X|)≥ c) ≤ E(|X|)/c .(b) Let Y≥ 0 be such that E(Y)= 0 . Show P(Y =0) =1 .(c) Let A 1, A 2, ...be a monotone sequence of events with limit A. ShowP(A n) → P(A) as n→ ∞ .3.(a) Let X have pgf G(s)= .3+.5s+.2s 2 . Calculate Var(X) .(b) For the rv X in 3(a) sketch the df F(x)=P(X≤x) for -1

STA257- Test #1Time=2hrsNote: No aids. All questions are of equal value. Please sign this sheetwith your student # and hand it in with your test booklets.1.(a) Let X be a rv and c>0 a constant. Show P(|X|)≥ c) ≤ E(|X|)/c .(b) Suppose E(|Y|)= 0 . Show P(Y =0) =1 .(c) Let A 1, A 2, ...be a monotone sequence of events with limit A. ShowP(A n) → P(A) as n→ ∞ .2.(a) Let A and B be independent events. Show A and B c are independent.(b) Show P(A∪B∪C) ≤ P(A) + P(B) + P(C) .(c) Let A 1, A 2, ... be events with P(A i) = 1 , i=1,2, ... Show∞P( ∩Ai ) = 1 , n ≥ 2i=13.(a) Let X have pgf G(s)= .3+.5s+.2s 2 . Calculate E(X) .(b) For the rv X in 3(a) sketch the df F(x)=P(X≤x) for -1

STA257- Test #2Time=2hrsNote: No aids. Each of the 12 problems is worth 9 and the maximumgrade is 100. Please sign this sheet with your student # andhand it in with your test booklets.1.(a) Let X have df F . Let x be any point and suppose x nis a decreasingsequence such that x n! x as n! " . Show F(x n) ! F(x) , as n ! " .(b) Let Y# 0 be such that E(Y)= 0 . Show P(Y =0) =1 .(c) Let A 1, A 2, ... be events with P(A i) = 1 , i=1,2, ... ShowP( $i=1"Ai) = 12. (a) Derive the mgf of a N(0,1) rv .(b) Let Z 1, Z 2be iid N(0,1) . Calculate the pdf of Y = Z 1 2 + Z 22.(c) Show that a sum of n independent normals is normal.3. (a) Let X 1,...,X nbe iid with mean µ and variance % 2 . SupposeX = (X 1+ ...+ X n)/n . Calculate the mean and variance of X .(b) Let Y = X 1+ X 2, where X 1& Poisson (2) and is independent of X 2. If Yis Poisson (5) show X 2& Poisson (3 ) .(c) Let U, V be iid Poisson (') . Calculate P(U=k | U+V= n ) , k=0 , ..., n .4.(a) Let Y = 1 + (X 1/X 2) , where X 1, X 2are iid N(0,1) . Derive the pdf ofY .(b) Let X 1, X 2be iid exponential (') . Derive the pdf of Y= X 1/(X 1+X 2) .(c) Let U, V, W be iid Poisson (3) . Set X=U+V and Y= V+W . Obtain thejoint pgf of X and Y .The EndInformationThe pdf of a N(0,1) is f(x) = (2() -1/2 exp(-x 2 / 2 )The Poisson(') probabilities are e -' ' k /k!The exponential(') has pdf f(x) ='exp(-'x) , x>0 and 0 ow .

STA257- Test #2Time=2hrsNote: No aids. Each of the 12 problems is worth 9 and the maximumgrade is 100. Please sign this sheet with your student # andhand it in with your test booklets.1.(a) Let X have df F . Let x be any point and suppose x nis a decreasingsequence such that x n! x as n! " . Show F(x n) ! F(x) , as n ! " .(b) Let Y# 0 be such that E(Y)= 0 . Show P(Y =0) =1 .(c) Let A 1, A 2, ... be events with P(A i) = 1 , i=1,2, ... ShowP( $i=1"Ai) = 12. (a) Derive the mgf of a N(µ,% 2 ) rv .(b) Let Z 1, Z 2be iid N(0,1) . Calculate the pdf of Y = Z 1 2 + Z 22.(c) Show that a sum of n independent Poisson’s is Poisson .3. (a) Let X 1,...,X nbe iid with mean µ and variance % 2 . SupposeX = (X 1+ ...+ X n)/n . Calculate the mean and variance of X .(b) Let Y = X 1+ X 2, where X 1& Poisson (1) and is independent of X 2. If Yis Poisson (5) show X 2& Poisson (4 ) .(c) Let U, V be iid Poisson (') . Calculate P(U=k | U+V= n ) , k=0 , ..., n .4.(a) Let Y = 1 - (X 1/X 2) , where X 1, X 2are iid N(0,1) . Derive the pdf ofY .(b) Let X 1, X 2be iid exponential (') . Derive the pdf of Y= X 1/(X 1+X 2) .(c) Let U, V, W be iid Poisson (2) . Set X=U+V and Y= V+W . Obtain thejoint pgf of X and Y .The EndInformationThe pdf of a N(0,1) is f(x) = (2() -1/2 exp(-x 2 / 2 )The Poisson(') probabilities are e -' ' k /k!The exponential(') has pdf f(x) ='exp(-'x) , x>0 and 0 ow .