Let X be a random variable with EX 1 and VarX 4 Let Y 2X

Let X be a random variable with E(X) = 1 and Var(X) = 4. Let Y = 2X + 7. Find E(Y) Find Var(Y) Find the standard deviation of X Find E(X^2) Find Cov(X, Y), where Cov denotes the covariance.

Solution

Since,

E(X) = 1

Var(X) = 4

and

Y = 2X + 7

Therefore,

(a) E(Y) = E(2X+7) = E(2X) + E(7) = 2E(X) + 7 = 2*1+7 = 2+7 = 9 Answer

(b) Var(Y) = Var(2X+7) = (2²)Var(X) = 4 * Var(X) = 4 * 4 = 16 Answer

(c) Standard Deviation of X = Square root of Var(X) = Root(4) = 2 Answer

(d) E(X²) ,

Since, Var(X) = E(X²) - E²(X)

=> 4 = E(X²) - 1²

=> 4 = E(X²) - 1

=> 4 + 1 = E(X²)

=> 5 = E(X²)

=> E(X²) = 5 Answer

XY = X*[2X+7] = 2X²+7X

(e) Cov(X,Y) = E(XY) - E(X)E(Y) = E(2X²+7X) - 1*9 (since E(X)=1,E(Y)=9)

   = 2E(X²) + 7E(X) - 9 = 2*5 + 7*1 - 9 = 10 + 7 - 9 = 8

therefore, Cov(X,Y) = 8 Answer