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- X1 and X2 are independent random variables such that Xi has PDF fXi(x)={λiexp(−λix) when x≥0, 0 otherwise}. What is P[X2<X1]?Let X, Y be two Bernoulli random variables anddenote by p = P (X = 1), q = P (Y = 1) and r = P (X = 1, Y = 1). Prove that X and Y are independent if and only if r = pq.Let X1 and X2 be independent random variables for which P(Xi = 1) = 2/5 and P(Xi = 2) = 3/5 . Define U = X1 + X2 and V = X1 x X2. Calculate Cor(U, V )
- Let X and Y be random variables. Suppose Var(X) = 1.6, Var(Y) = 1.2, and Cov(X, Y) = 0.6. Let Z = -1.2X + 1.9Y + 4.2. Calculate Var(Z).Let X1,X2,... be a sequence of identically distributed random variables with E|X1|<∞ and let Yn = n−1max1≤i≤n|Xi|. Show that limnE(Yn) = 0Suppose X and Y are random variables with E[XY ] = 6, E[Y ] = 4 and E[X] = 5 Find Cov(X; Y )
- 1. X1, X2, and X3 are i.i.d. random variables with each Xi ∼ N (0, σ2). Let Z and Y be defined asZ = αX1 + βX2Y = aX1 + bX2 + cX3where α, β, a, b, c are real valued nonzero constants. Find the distribution of Z|Y .Let X1, X2, X3, . . . be a sequence of independent Poisson distributed random variables with parameter 1. For n ≥ 1 let Sn = X1 + · · · + Xn. (a) Show that GXi(s) = es−1.(b) Deduce from part (a) that GSn(s) = ens−n.Let X, Y, and Z be jointly distributed random variables. Prove that Cov(X + Y, Z) = Cov(X, Z) + Cov(Y, Z).
- 2. Y1, Y2, ..., Yn are i.i.d. exponential random variables with E{Yi} = 1/θ. Find thedistribution of Y =1 nPiYi.Let Xi be arandom sample from U(0,1)prove that Xn’ convarges in probability to 0.50Let Xi and Yi be random variables with Var(Xi) = σx2 and Var(Yi) = σy2 for all i ∈ {1, . . . , n}. Assume that each pair (Xi, Yi) has correlation Corr(Xi, Yi) = ρ, but that (Xi,Yi) and (Xj,Yj) are independent for all i ̸= j. (a) What is Cov(Xi,Yi) in terms of σx, σy and ρ? (b) Show that Cov(Xi,Y ̄) = (ρσxσy)/n, where Y ̄ is the average of the Yi (c) Determine Cov(X ̄,Y ̄). B2. Consider the random variables Xi and Yi from question B1 again. (a) Show that the sample covariance is an unbiased estimator of Cov(X1,Y1). Hint: consider the equality Xi − X ̄ = (Xi − μ) − (X ̄ − μ). (b) Can you conclude from the statement in part (a) that the sample correlation is an unbiased estimator of Corr(X1, Y1)? Justify your answer.