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Homework problems for MATH 5075/6820 Deadline 29 January Problem 1 Let X and Y be independent and identically distributed exponential random variables with EX = EY = 1. Compute the distribution function of Z = X + Y . Solution. By the convolution formula, h(t) = ∫ ∞ −∞ f(t− u)f(u)du = ∫ ∞ 0 f(t− u)e−udu = ∫ t 0 e−(t−u)e−udu = te−t if t > 0 and the density is 0 if t ≥ 0. The exponential density is f(t) = e−tI{t ≥ 0}. Problem 2∗ Let X, Y and Z be independent and identically distributed exponential random variables with EX = EY = EZ = 1. Compute the distribution function of V = ZX+Y . Solution. The density of Z is exponential f(t) = e−tI{t ≥ 0} and the density of X + Y is Gamma(1) g(t) = te−tI{t ≥ 0}. Now x > 0 P {V ≤ x} = ∫ ∞ 0 P { Z t ≤ x } g(t)dt = ∫ ∞ 0 P{Z ≤ tx}te−tdt = ∫ ∞ 0 (1− e−tx)te−tdt and ∫ ∞ 0 (1− e−tx)te−tdt = ∫ ∞ 0 te−tdt− ∫ ∞ 0 te−t(x+1)dt∫ ∞ 0 te−tdt = 1 (the integral of a density is 1) Using the density of a Gamma(1, 1/(x+ 1)) and the notation λ = 1/(1 + x)∫ ∞ 0 te−t(x+1)dt = 1 1 + x ∫ ∞ 0 t λ e−t/λdt = 1 1 + x The derivative is the density fV (x) = 1 (1 + x)2 I{x > 0}. Problem 3 Let X1, X2, . . . , Xn be independent and identically distributed random variables with distribution function F (x) = 0, x < 0="" 1−="" 1="" 1="" +="" x="" ,="" x="" ≥="" 0.="" let="" xn,n="max1≤i≤nXi" and="" yn="Xn,n/n." show="" that="" yn="" converges="" in="" distribution="" and="" determine="" the="" limit.="" solution.="" p="" {yn="" ≤="" x}="P" {="" max="" 1≤i≤n="" xi="" ≤="" xn="" }="Fn(xn)" 2="" and="" for="" x=""> 0 Fn(xn) = ( 1− 1 1 + nx )n = ( nx 1 + nx )n = ( 1 + nx nx )−n = ( 1 + 1/x n )−n = [( 1 + 1/x n )n]−1 → e−1/x The limit distribution is 0 if x < 0.="" we="" used="" the="" result(="" 1="" +="" y="" n="" )n="" →="" ey="" for="" all="" y="" problem="" 4∗="" let="" x1,="" x2,="" .="" .="" .="" ,="" xn="" be="" independent="" and="" identically="" distributed="" random="" variables="" with="" distribution="" function="" f="" (x)="" 0,="" x="">< 0="" 1−="" 1="" 1="" +="" x="" ,="" x="" ≥="" 0.="" let="" fn(t)="" be="" the="" distribution="" function="" of="" yn="max1≤i≤nXi/n." show="" that="" there="" is="" a="" distribution="" function="" g="" and="" a="" constant="" c="" such="" that="" sup=""><∞ |fn(x)−g(x)|="" ≤="" c="" n="" ,="" where="" g="" denotes="" the="" limiting="" distribution="" function.="" s="" ¯="" olution.="" we="" use="" the="" distribution="" of="" yn="" from="" problem="" 3.="" we="" also="" use="" |="" log(1="" +="" h)−="" h|="" ≤="" h2/4,="" if="" |h|="" ≤="" h0="" and="" ∣∣∣e−x="" −="" e−x+h∣∣∣="" ≤="" |h|="" for="" all="" x=""> 0. Now ( 1 + 1/x n )−n = exp ( −n log(1 + 1/x n ) ) and since we can assume 1/(nx) ≤ 1/n ≤ h0 exp (−1/x− 1/(4n)) ≤ exp ( −n log ( 1 + 1/x n )) ≤ exp (−1/x+ 1/(4n)) and ∣∣∣exp (−1/x− 1/(4n))− e−1/x∣∣∣ ≤ 1 4n ,∣∣∣exp (−1/x+ 1/(4n))− e−1/x∣∣∣ ≤ 1 4n . Deadline 5 February Problem 5 Let X1, X2, . . . , XN be independent and identically distributed random vectors with EXi = 0 and EX 2 i = 1. Compute E ( n∑ i=1 1 i Xi )2 and show ∣∣∣∣∣ n∑ i=1 1 i Xi ∣∣∣∣∣ = OP (1). Problem 6∗. Let (Xi, Yi), 1 ≤ i ≤ n be independent and identically distributed random variables 3 with EXi = EYi = 0, EX 2 i = σ 2 1, EY 2 i = σ 2 2 and corr(Xi, Yi) = ρ. Compute E ( n∑ i=1 Zi )2 , where Zi = Xi + 1 i Yi, 1 ≤ i ≤ n. Problem 7. Let X1, X2, . . . , Xn be a sequence of random variables with EXi = 0, EXiXj = σ2, i = j, ρ |i− j| = 1, 0, |i− j| > 1. Compute E ( n∑ i=1 Xi )2 and show that 1 n n∑ i=1 Xi P→ 0. Problem 8∗. Let X1, X2, . . . , Xn be independent and identically distributed exponential random variables with EXi = 1. Compute E ( 1 X1 +X2 + . . .+Xn ) . Deadline 12 February Problem 9 Let �i be a sequence of random variables with E�i = 0, E� 2 i = σ 2 and E�i�j = 0 if i 6= j. Let x0 = 0 and |ρ| > 1. The sequence xk defined by xi = ρxi−1 + �i, i = 1, 2, . . . . Compute Exk and Exkxk+2. Problem 10 Let �i,−∞ < i="">< ∞,="" be="" independent="" and="" identically="" distributed="" random="" variables="" with="" �i="0" and="" �="" 2="" i="σ" 2.="" let="" xi="" be="" the="" stationary="" solution="" of="" xi="1" 3="" xi−1="" +="" �i,="" −∞="">< i=""><∞. if="" var(x0)="100," determine="" σ="" 2.="" problem="" 11*="" let="" �i,−∞="">< i="">< ∞,="" be="" independent="" and="" identically="" distributed="" normal="" random="" variables="" with="" �i="0" and="" �="" 2="" i="σ" 2.="" let="" xi="" be="" the="" stationary="" solution="" of="" xi="ρxi−1" +="" �i,="" −∞="">< i=""><∞, |ρ|="">< 1.="" compute="" the="" joint="" distribution="" of="" xk="" and="" x`.="" 4="" problem="" 12*="" let="" �i,−∞="">< i="">< ∞,="" be="" independent="" and="" identically="" distributed="" random="" variables="" with="" �i="0" and="" �="" 2="" i="σ" 2="" and="" define="" x0="∞∑" `="0" c`�−`.="" prove="" that="" x0="" is="" well="" defined.="" i.e.="" the="" infinite="" sum="" defining="" x0="" converges="" with="" probability="" 1="" if="" ∞∑="" `="0" c2`=""><∞. deadline="" 19="" february="" problem="" 13="" let="" �i,−∞="">< i="">< ∞="" be="" independent="" and="" identically="" distributed="" random="" variables="" with="" e�i="0" and="" e�="" 2="" i="σ" 2.="" write="" the="" stationary="" solution="" of="" xi="0.7xi−1" −="" 0.10x2="" +="" �i="" in="" a="" causal="" for.="" i.e.="" an="" infinite="" sum="" of="" the="" {�j="" ,="" j="" ≤="" i}.="" problem="" 14="" let="" �i,−∞="">< i="">< ∞="" be="" independent="" and="" identically="" distributed="" random="" variables="" with="" e�i="0" and="" e�="" 2="" i="σ" 2.="" write="" the="" stationary="" solution="" of="" xi="−0.25x2" +="" �i="" in="" a="" causal="" for.="" i.e.="" an="" infinite="" sum="" of="" the="" {�j="" ,="" j="" ≤="" i}.="" problem="" 15*="" let="" �i,−∞="">< i="">< ∞="" be="" independent="" and="" identically="" distributed="" random="" variables.="" let="" xi="" be="" the="" stationary="" causal="" solution="" of="" xi="φ1xi−1" +="" φ2xi−2="" +="" .="" .="" .+="" φpxi−p="" +="" �i.="" prove="" that="" e|xi|ν=""><∞ if="" and="" only="" if="" e|�0|ν=""><∞, where="" ν=""> 0. Problem 16* Let �i,−∞ < i="">< ∞="" be="" independent="" and="" identically="" distributed="" random="" variables,="" e�i="0" and="" e�="" 2="" i="σ" 2.="" show="" that,="" if="" |φ1|+="" |φ2|="">< 1, then xi = φ1xi−1 + φ2xi−2 + �i has a stationary causal solution. 1,="" then="" xi="φ1xi−1" +="" φ2xi−2="" +="" �i="" has="" a="" stationary="" causal="">