r any two random variables X and Y, Var(X) ≥ E[Var(X|Y)]. TRUE FALSE
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O TRUE
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- Suppose that X, Y, and Z are jointly distributed random variables, that is, they are defined on the same sample space. Suppose that we also have the following. E(X)=-6 E(Y)=-9 E(Z)=3 Var (X)=8 Var (Y)=14 Var (Z)=25 Compute the values of the expressions below. E (-1- 37) - I -2x - 5z 4 Var (47) - 2 - I e(32°) - [I DloLet X and Y be two random variables. Suppose we know E(X) = 7 and E(Y)=3.Let Z = X − Y be a random variable. What is E(Z)? Justify your answer.Random variable X is independent of random variable Y, and X-B(10, 0.3), Y-B(20, 0.3), compute P(X=1 | X+Y=4).
- Let X1, X2, and.X3 be independent and normally distributed random variables with E(X1) 4, E(X2) = 3, E(X3) = 2, Var(X1) = 1, Var(X2) = 5, Var(X3) = 2. Let Y = 2X1 + X2 – 3X3. Find 2. the distribution of Y.IfXand Y are Two Random Variables and X, Y and o, 2 are the Corresponding Means and Variance of X and y Y respectively then, Х-Х Ү-Ү Y -Y 1 x Cov(X, Y) Cov Oy Ox Oya) Given an exponential random variable X with lambda L, what is the density at X=2. Answer in terms of L. b) Using the random variable defined in the previous question, let L be 5. Find the probability of P(X<10).
- Let X be a random variable. Which of the following statement is INCORRECT? If V (X) = 0, then there exists an x, such that Pr(X = x) = 1. There are cases that V(X) does not exist. By the definition of the random variable, the range of X is a subset of R; therefore, there must exist at least one x such that Pr(X = x) > 0. If Pr(X = 1) (E(X))².Let X be a random variable and c be a constant. Then Var(X+c) = Var(X), always. True Falselet x be a discrete random variable with probability function p(x)= cx/20 for x = 2,3,4,5,6,8,12 what is the value of c that makes p(x) a valid probability function?
- Suppose a and b be two possible values of a random variable X with a > b. The probability that X lies between a and b is P(a > X > b) = F (a) - F (b) Select one: O True O FalseLet X and Y denote two random variables. Which of the following can be used to compute Var(X)? A. E[Var(X|Y)] + Var(Var(X|Y)) B. E[E[X|Y]] + Var(Var(X|Y)) C. E[Var(X|Y)] + Var(E[X|Y]) D. Var(E[X|Y]) + Var(Var(X|Y))ans Theorem 9.3. Let X and Y be independent random variables with finite variances, and a, b ER. Then Var(aX) = a²Var (X), Var (X+Y) = VarX + Var Y, Var (ax + bY) = a²VarX + b²Var Y. sercise Prove the theorem. Remark 9.2. Independence is sufficient for the variance of the sum to be equal to the sum of the variances, but not necessary. Remark 9.3. Linearity should not hold, since variance is a quadratic quantity. Remark 9.4. Note, in particular, that Var (-X) = Var(X). This is as expected, since switching the sign should not alter the spread of the distribution.
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