3. Let (X,Y) be a bivariate random variable, and let a and b be real constants. Show that (a) Cov(aX, bY) = ab Cov(X,Y). (b) Cov(X +a,Y + b) = Cov(X, Y). (c) Cov(X,aX + b) = aVar(X).
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- For the random variables X,Y Cov(X,Y) = -0.9 if Z=3-X then what is Cov(Z,Y)=???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.Suppose that the random variables X1,...,Xn form a random sample of size n from the uniform distribution on the interval [0, 1]. Let Y1 = min{X1,. . .,Xn}, and let Yn = max{X1,...,Xn}. Find E(Y1) and E(Yn).
- Suppose that Z1, Z2, . . . , Zn are statistically independent random variables. Define Y as the sum of squares of these random variablesSuppose we have two random variables X and Y. For a, b > 0, a2+b2=13. Var(aX) = 16 Var(bY) = 36 Var(X - Y) = 2 Cov(2X, Y) = 6 1) What is ab? 2) What is Corr(X, Y)?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.
- If X is a continuous random variable with X ∼ Uniform([0, 2]), what is E[X^3]?Suppose that three random variables X1, X2, X3 form a random sample from the uniform distribution on interval [0, 1]. Determine the value of E[(X1-2X2+X3)2]Let x and y be random variable such that the mean and variance of X are 2 and 4. respectively, while the mean and variance of y are 6 and k, respectively. A sample of size 4 is taken from the x-distribution and a sample of size 9 is taken from the y-distribution .If p[(x-y)>8]=0.0228,then what is the value of the constant k?
- Let X and Y be random variables. Suppose Var(X) = 1.6, Var(Y) = 1.5, and Cov(X, Y) = 0.1. Let Z = -1.2X + 1.2Y + 4.7. Calculate Var(Z).A poisson random variables has f(x,3)= 3x e-3÷x! ,x= 0,1.......,∞. find the probabilities for X=0 1 2 3 4 and also find mean and variance from f(x,3).?Let X and Y be two random variables and let r, s, t, and u be real numbers. a. Show that Cov(X+s, Y+u) = Cov(X,Y)b. Show that Cov(rX, tY) = rtCov(X,Y)c. Show that Cov(rX +s, tY + u) = rtCov(X,Y)