We have the following information about the random variables X and Y: 4 8 11 4 5 ox = o Cov(X, Y) 225 15 75 225 Calculate the standard deviation of Z 3Y — Х. oz

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 ?x and ?y, variances ?2x and ?2y, and population correlation coefficient ? (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 ?x?y? is called the covariance of x and y. If x and y are independent, then ? = 0 and the formula for ?2w 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…

M 7 Two statistically independent random variables X and Y have mean values as 2 and 4 respectively. They have second moments as 8 and 25 respectively. Find the mean, second moment and variance of the random variable o = 3X – Y.

Let X1, X2, and X3 be uncorrelated RVs each with the same mean, µ, and variance, o?. Find expressions for the following in terms of the mean and variance: 1. Cov(X1 + X2, X, + X3). 2. Cov(X1 + X2, X – X2).

Consider the following joint random variable pmf for X and Y: x y   -1 0 1/4 0 1 1/8 1 1 1/2 1 0 1/8   What is the correlation of X and Y?

4. Let X - N(u, o²). The characteristic function (ch.f.) of X is given by $(w) = ejHw-o?w?/2, W E R, where j = v-1 is the imaginary unit. Use the ch.f. to find the mean and variance of X.

Bapu

Let Z1 and Z2 be independent standard normal N(0, 1) random variables. Let X = Z1 + 2Z2, and let Y = 2Z1 + Z2. a. Find rho( X, Y ) = correlation of X and Y. b. Find the numerical value of P{ X > 0 and Y > 0 }. HINT: Use trigonometry.