X1, X2, and X3 all have a normal distribution with mean 1 and variance 1. What...
Let X1 be a normal random variable with mean 2 and variance 3, and let X2 be a normal random variable with mean 1 and variance 4. Assume that X1 and X2 are independent. What is the distribution of the linear combination Y = 2X1 + 3X2?
x={x1,x2,x3} has the 3-variate normal dustribution with mean 0 and variance covariance matrix=(3 1 1 1 3 1 1 1 4) find PDF of x in full
1. Suppose that X1, X2, and X3 E(X1) = 0, E(X2) = 1, E(X3) = 1, Var(X1) = 1, Var(X2) = 2, Var(X3) = 3, Cov(X1, X2) = -1, Cov(X2, X3) = 1, where X1 and X3 are independent. a.) Find the covariance cov(X1 + X2, X1 - X3). b.) Define U = 2X1 - X2 + X3. Find the mean and variance of U.
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...
Let X1, X2, X3, X4 be a random sample from a standard normal population. What is the probability distribution (give the name of the distribution and the value of any parameter(s)) of (a). (X1 - Xbar)^2 + (X2 - Xbar)^2 + (X3 - Xbar)^2 + (X4 - Xbar)^2 (b). ((X1 + X2 + X3 + X4)^2)/4
X1, X2, X3, X4,X5,X6,X7,X8 are independent identically distributed random variables. Their common distribution is normal with mean 0 and variance 4. Let W = X12+ X22 + X32 + X42+X52+X62+X72+X82 . Calculate Pr(W > 2)
4 persons each have playing cards, with person 1 having X1 cards, person 2 having X2 cards, person three having X3 cards and person 4 having X4 cards. The random variables of X1, X2, X3 and X4 are all independent and normally distributed with mean = 100 and SD = 24. 1) If Y = X1+X2+X3+X4 (all the cards of alle the persons), what is then E(Y)? What is Var(Y)? 2) Denoting A as the average no. of cards per...
Suppose you have a random sample {X1, X2, X3} of size n = 3. Consider the following three possible estimators for the population mean u and variance o2 Дi 3D (X1+ X2+ X3)/3 Ti2X1/4 X2/2 X3/4 Дз — (Х+ X,+ X3)/4 (a) What is the bias associated with each estimator? (b) What is the variance associated with each estimator? (c) Does the fact that Var(i3) < Var(1) contradict the statement that X is the minimum variance unbiased estimator? Why or...
(1 point) A normal distribution with mean and variance o is independently sampled three times, yielding values x1, x2, and X3. Consider the three estimators û = X1 + 5x2 A2 = x - x2 + x3, and Find the expected value of each estimator (type mu for and sigma foro): E) EG) E) = Which estimator(s) are biased and which are unbiased? Estimator : ? Estimator 2: ? Estimators: ? Find the variance of each estimator (type mu for...
= = 3, Cov(X1, X2) = 2, Cov(X2, X3) = -2, Let Var(X1) = Var(X3) = 2, Var(X2) Cov(X1, X3) = -1. i) Suppose Y1 = X1 - X2. Find Var(Y1). ii) Suppose Y2 = X1 – 2X2 – X3. Find Var(Y2) and Cov(Yı, Y2). Assuming that (X1, X2, X3) are multivariate normal, with mean 0 and covariances as specified above, find the joint density function fxı,Y,(y1, y2). iii) Suppose Y3 = X1 + X2 + X3. Compute the covariance...