Properties of Expectation and Variance Suppose we have two independent discrete random variables, say X1 and X2. Suppose further E(X1) = 21 Var(X1) = 126 and E(X2) = 3.36 Var(X2) = 1.38
Compute the Expectations and Variances of the following linear combinations of X1 and X2.
a) E(πX1 + eX2 + 17)
b) E(X1 · 3X2)
c) Var( (√ 13X2) + 46)
d) Var(X1 + 2X2 + 14)
Properties of Expectation and Variance Suppose we have two independent discrete random variables, say X1 and X2. Suppose...
Let X1, X2, X3 be independent random variables with E(X1) = 1, E(X2) = 2 and E(X3) = 3. Let Y = 3X1 − 2X2 + X3. Find E(Y ), Var(Y ) in the following examples. X1, X2, X3 are Poisson. [Recall that the variance of Poisson(λ) is λ.] X1, X2, X3 are normal, with respective variances σ12 = 1, σ2 = 3, σ32 = 5. Find P(0 ≤ Y ≤ 5). [Recall that any linear combination of independent normal...
3. Suppose that X1, X2, , Xn are independent random variables with the same expectation μ and the same variance σ2. Let X--ΣΑι Xi. Find the expectation and variance of
Consider the independent random variables X1, X2, and X3 with - E(X1)=1, Var(X1)=4 - E(X2)=2, SD(X2)=3 - E(X3)=−1, SD(X3)=5 (a) Calculate E(5X1+2). (b) Calculate E(3X1−2X2+X3). (c) Calculate Var(5X1−2X2).
Suppose X1, X2, . . . are independent discrete random variables,
having the same distribution, and E[Xi] > 0, for each i. Is thus
true for any two positive integers n and m?:
Why not, or why yes?
6. Suppose random variables X1, X2, X3 have the following properties: E(X1) = 1; E(X2) = 2; E(X3) = −1 V(X1) = 1; V(X2) = 3; V(X3) = 5 COV (X1,X2) = 7; COV (X1,X3) = −4; COV (X2,X3) = 2 Let U = X1 −2X2 + X3 and W = 3X1 + X2. (a) Find V(U) (b) Find COV (U,W).
1 [3]. Let X1,X2, X3 be iid random variables with the common mean --1 2-4 and variance σ Find (a) E (2X1 - 3X2 + 4X3); (b) Var(2X1 -4X2); (c) Cov(Xi - X2, X1 +2X2).
Let X1 and X2 be independent random variables with mean μ and variance σ2. Suppose we have two estimators 1 (1) Are both estimators unbiased estimatros for θ? (2) Which is a better estimator?
Let X1 and X2 be independent random variables with means μ1 and μ2, and variances σ21 and σ22, respectively. Find the correlation of X1 and X1 + X2. Note that: The covariance of random variables X; Y is dened by Cov(X; Y ) = E[(X - E(X))(Y - E(Y ))]. The correlation of X; Y is dened by Corr(X; Y ) =Cov(X; Y ) / √ Var(X)Var(Y )
8. We say that two discrete random variables X and Y , are independent when P(X = a, Y = b) = P(X = a)P(Y = b) for all a and b in the corresponding sample spaces. Let Xị and X, be independent Poisson random variables with parameters l1 = 3 and dy = 2 respectively. Find the probability of the event that X1 + X2 = 3. Hint: Since {X1 + X2 = 3} = {X} = 0, X2...
Let X1 and X2 be random variables, not necessarily independent. Show that E [X1 + X2] = E [X1] + E [X2]. You may assume that X1 and X2 are discrete with a joint probability mass function for this problem, while the above inequality is true also for continuous random variables.