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- Let X be a random variable such that E(X²) a. V(3X-3) b. E(3x-3) = 58 and V(X) = 40. ComputeLet X be a random variable and a real number. Show that E(X - a)² = varX + (µ − a)² Hereμ = EX is the expected value of the random variable X and varX = E(X - μ)^2 is the variance of the random variable X. Guidance: start from the representation - (X-a)^2 = (X µ + μ- a)^2 and group the right side of the representation appropriately into the form (Z + b)^2, where Z is some random variable and b is a real number and open the square. The task should be solved with the help of the expected value calculation rules.The random variable x is N(10,1). Find f(x(x-10)²<4).
- Let x1 and x1 be independent standard normal random variables. If Y1= x1 + 2x2 and Y2 = 2x1 - x2, find Fy1,y2(1,2)Let X be a binomially distributed random variable B(n, p) with E(X) = 2 and Var(X) = 4/3. Find n.show all solution 1. Suppose X is a random variable with E(X2)=10 and E(X)=2. Find the following: a) E(2X + 5) b) Var(X) c) Var(2X + 5) d) Var(2X – 5)
- Let Y₁ = (x,-X₂), where X, and X₂ are independent random variables and each of them X, 2 is distributed as x² (2). Find the p.d.f. of Y₁.Suppose Xis a random variable with E(X) = 2, and E(X2) = 5. Compute the following quantities: a) Var(X) b) E(X−1)2 c) Var(−X−3) d) E(5X+ 4)Suppose that X and Y are random variables with E(X) 2 , E(Y) = 5 and E(x?) = 8, E(Y?) = 30 and cov(2X – Y,4X – 3Y) = -12, then cov(-5X – 3,2Y + 4) is equal to: Hint: cov(ax + bY,cX + dY) = acV(X) + bdV(Y) + (bc + ad)cov(X,Y) O -120 O -59 O -64 O -30
- X and Y are discrete random variables. If var(X)= var(Y) = o², cov(X,Y)=, find var(2X-3Y)Suppose that X andY are random variables with E(X) = 2 , E(Y) = 5 and E(X²) = 8, E(Y?) = 30 and cov(2X – Y,4X – 3Y) = -12, then cov(-5X – 3, 2Y + 4) is equal to: Hint: cov(aX + br, сх + dr) %3 асV(х) + bdv (Y) + (be + ad)cov(X,Y) -64 -30 -120 -59Suppose that X,Y are both standard normal random variables and Cov(X,Y)= 1 (a) Compute E (X² – Y²)| (b) Is it possible to compute P (X +Y > 3) from the given information? If so compute, otherwise explain.
