Let X1, X2, ..., X, be n random variables, then (i a; X.) = £ a?V (X) + 2 _£ E a;a; E a? V (X;) + 2 £ d; Aj Cov (X¡, X;) V i=1 i = 1 i =1 j=1
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- Show that if X, Y are independent random variables, then Cov(X, Y ) = 0.Suppose that Z1, Z2, . . . , Zn are statistically independent random variables. Define Y as the sum of squares of these random variablesSuppose the random variable y is a function of several independent random variables, say x1,x2,...,xn. On first order approximation, which of the following is TRUE in general?
- Let X1, X2, ..., Xn be independent and identically distributed random variables, X ∼ Exp(λ). Show that the sum X1 + X2 + · · · + Xn is Γ(n, λ) distributed.If for a pair of random variables X,Y , their covariance is zero: Cov(X, Y ) = 0. Can we say that X and Y are independent? If so, please prove it. If not, please provide a counter-example.Let X1,...,Xn be iid random variables with expected value 0, variance 1, and covariance Cov [Xi,Xj] = ρ, for i≠j. Use Theorem of linearity of expectation to find the expected value and variance of the sum Y = X1 +...+Xn.
- Let X1, X2, X3 and X4 be exponential(1) random variables. Find the joint distribution of X1/(X1+X2+X3+X4) and (X1+X2)/(X1+X2+X3+X4) and (X1+X2+X3)/(X1+X2+X3+X4) using Jacobian method?Let X and Y be independent random variables such that Var[X] = 4.8 and Var[Y ] = 9.7. [a] How can we prove that 2X and 3Y are independent? Please show your proof.Suppose that random variables X and Y are defined on a sample space with only two elements. Suppose that Cov(X, Y ) = 0. Prove that X and Y are independent.
- Let � be a random variabale satisfying �(0,1).Without using the normalcdf(...) function, find �(-3≤�<-2).Assume Z1, Z2, . . . , Zn are independent standard normal random variables. The random variable Y defined byLet X, Y be two Bernoulli random variables and denote 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 {Xi, Yi}ni=1 be a sample of n i.i.d. copies of (X, Y ). Based on this sample, we want to test whether X and Y are independent, i.e., whether r = pq Define ˆp = sum(Yi)/n, ˆq =sum(Yi)/n and ˆr =sum(XiYi)/n- Prove that these are, respectively, consistent estimators of p, qand r.– Show that the vector (ˆp, q ˆ, rˆ) is asymptotically normal and find the asymptotic covariance matrix.– Using the previous question combined with the Delta-method, prove that√n ((ˆr − pˆqˆ) − (r − pq)) → t, Vas n → ∞ in distribution, where V depends on p, q and r.– Consider the following hypotheses:H0: “X and Y are independent” vs H1: “X and Y are not independent”Assuming that H0 is true, show that V = pq (1 − p) (1 − q) and propose a consistent estimator of V .– Using the last two questions, propose a test with asymptotic level α ∈ (0,1).