probability course 01) 6 and Let X and Y be two independent random variables. Suppose that...
9. Let X and Y be two random variables. Suppose that σ = 4, and σ -9. If we know that the two random variables Z-2X?Y and W = X + Y are independent, find Cov(X, Y) and ρ(X,Y). 10. Let X and Y be bivariate normal random variables with parameters μェー0, σ, 1,Hy- 1, ơv = 2, and ρ = _ .5. Find P(X + 2Y < 3) . Find Cov(X-Y, X + 2Y) 11. Let X and Y...
(Sums of normal random variables) Let X be independent random variables where XN N(2,5) and Y ~ N(5,9) (we use the notation N (?, ?. ) ). Let W 3X-2Y + 1. (a) Compute E(W) and Var(W) (b) It is known that the sum of independent normal distributions is n Compute P(W 6)
please show steps, thank you (Sec. 5.2, 00) Suppose X and Y are independent random variables with E[X] = 6, E[Y ] = −3, Var[X] = 9, and Var[Y ] = 25. Find: (a) E[2Y − X] (b) Var[2Y − X] (c) Cov[X, Y ] (d) ρ[X, Y ] (e) Cov[5X + Y, Y ] (f) Cov[X, 2Y − X]
Problem D: Suppose X1, .,X, are independent random variables. Let Y be their sum, that is Y 1Xi Find/prove the mgf of Y and find E(Y), Var(Y), and P (8 Y 9) if a) X1,.,X4 are Poisson random variables with means 5, 1,4, and 2, respectively. b) [separately from part a)] X,., X4 are Geometric random variables with p 3/4. i=1
6. Let the random variables X and Y represent the population of two species/organisms that compete with each other for survival. Suppose that the probability density function p(x, y) of these random variables is proportional to rye-(az+By) (a) What would be an appropriate sample space S for the random variables? Justify your choice of S. (b) Let p(x, y)-Krye-(az-+8v). What should be the proportionality constant K? (c) What is the most probable set of populations (X, Y)? (Hint: The probability...
Suppose XX and YY are independent random variables for which Var(X)=7Var(X)=7 and Var(Y)=7.Var(Y)=7. (a) Find Var(X−Y+1).Var(X−Y+1). (b) Find Var(2X−3Y)Var(2X−3Y) (c) Let W=2X−3Y.W=2X−3Y. Find the standard deviaton of W.W.
Let X and Y be independent random variables with pdf 2-y , 0sys2 2 f(x) 0, otherwise 0, otherwise ) Find E(XY) b) Find Var (2X+3Y)
Let X and Y be two independent random variables such that E(X) = E(Y) = u but og and Oy are unequal. We define another random variable Z as the weighted average of the random variables X and Y, as Z = 0X + (1 - 0)Y where 0 is a scalar and 0 = 0 < 1. 1. Find the expected value of Z , E(Z), as a function of u . 2. Find in terms of Oy and...
Let X, Y be independent random variables with E[X] = E[Y] = 0 and ox = Oy = 5. Then Var(2x+3Y) = 1. True False
Let X, Y be independent random variables with E[X] = E[Y] = 0 and ox = oy = 5. Then Var(2x +3Y) = 1. True False