iv) E[E[Y²|X]]_ (EY)²
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Solve subpart (iv). i-iii already solved
![Let X, Y be positive random variables (not necessarily independent). Assuming
X, Y have all finite moments, write and explain briefly the most appropriate of ≤, ≥, =, ? in
the blank, where '?' means no relationship holds in general.
i) E(X³). √E(X²) E(X4)
ii) P(|X+Y> 2)_ -¹/E((X+Y) ¹)
iii) E[√x +3].
iv) E[E[Y²|X]].
16
√E[(X+3)]
(EY)²](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F941b45b2-8ed9-4961-a892-5043ebafa0e8%2F22136833-d98f-4d56-97dc-4d6b2a9d58d0%2Fc2gzm5_processed.png&w=3840&q=75)
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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 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.X and Y are continuous random variables with pdf f(x,y) = 2x for0 ≤x ≤y ≤1, and f(x,y) = 0 otherwise. Find the conditional expectation ofY given X = x.