1. Cov(aX, Y) = aCov(X, Y). 2. Cov(X + Z,Y) = Cov(X, Y) + Cov(Z,Y). 3. Cov(aX+bZ,Y)= aCov(X,Y) +bCov(Z,Y).
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![1. Cov(aX,Y)=aCov(X, Y).
2. Cov(X + Z,Y)= Cov(X, Y) + Cov(Z,Y).
3. Cov(aX+bZ,Y)= aCov(X, Y) +bCov(Z,Y).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb01e0559-1b03-4a1c-ac9c-ab3a35fb5acd%2F926ae406-f635-4834-a09e-1a139e29b8eb%2Fa68rqjw_processed.png&w=3840&q=75)
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- The random variable Z is defined as a function of the random variables X and Y as follows: Z = 1 Moreover, it is known that E[X|Z] = Z² and that Z is zero mean. Compute E[Y|Z = 2] Y+1 and E[Y].Let X1 and X2 be independent random variables for which P(Xi = 1) = 2/5 and P(Xi = 2) = 3/5 . Define U = X1 + X2 and V = X1 x X2. Calculate Cor(U, V )Let X, Y, and Z be random variables, and let Cov(⋅,⋅) denote the covariance operator as usual. Suppose that the variance of X is 0.7, Cov(X,Y) = 0.4, Cov(X,Z) = 1.2, and Cov(Y,Z) = 0.8. Find each of the following to two decimal places. (a) Cov(3Y, 3X) (b) Cov(3Y + 3, 3X + 8Z)
- Let X1 and X2be independent exponential random variables: fX1(x1) = e−x1 and fX2(x2) = e−x2 1Suppose the discrete random variables X and Y are jointly distributed according to the given table. a. Compute the expected values E(X) and E(Y ), the variances V (X) and V (Y ), and also the covariance Cov(X, Y ) of X and Y. b. Determine whether X and Y are independent. c. Let T = X − Y . Compute E(T) and V (T)Let 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))
- The random variable Xi , i = 1, 2, models the proportion of type i switches on a panel that are turned off during a training exercise. The joint probability density function of X1 and X2 is (attached): Find the Covariance of (X1 , X2).Previously, you studied linear combinations of independent random variables. What happens if the variables are not independent? A lot of mathematics can be used to prove the following: Let x and y be random variables with means μx and μy, variances σ2x and σ2y, and population correlation coefficient ρ (the Greek letter rho). Let a and b be any constants and let w = ax + by for the following formula. μw = aμx + bμyσ2w = a2σ2x + b2σ2y + 2abσxσyρ In this formula, r is the population correlation coefficient, theoretically computed using the population of all (x, y) data pairs. The expression σxσyρ is called the covariance of x and y. If x and y are independent, then ρ = 0 and the formula for σ2w reduces to the appropriate formula for independent variables. In most real-world applications the population parameters are not known, so we use sample estimates with the understanding that our conclusions are also estimates.Do you have to be rich to invest in bonds and real estate? No, mutual fund…Let X have a mean of 13 and a variance of 1.1. Let Y have a mean of 9.1 and a variance of 0.75. The covariance of x and y is 0.35. Let Z=4X – 5Y+2. What is the mean and variance of Z?
- Let X and Y be independent and N(0, 1) distributed random variables. Let U = X and V = X/Y. Show that the random variable V is Cauchy distributed and find E(V ).Let pX(x) be the pmf of a random variable X. Find the cdf F(x) of X and sketch its graph along with that of pX(x) if pX(x)=1/3,x=−1,0,1, zero elsewhere
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