If E[X] = 1 and Var(X) = 5, (a) Find E[X^2]. (b)Derive the expression for Var(ax

Question

If E[X] = 1 and Var(X) = 5, (a) Find E[X^2]. (b)Derive the expression for Var(ax + b) where a and b is constant. c) Find Var(2x+4) and E[2x+4].
If E[X] = 1 and Var(X) = 5, (a) Find E[X^2]. (b)Derive the expression for Var(ax + b) where a and b is constant. c) Find Var(2x+4) and E[2x+4].
If E[X] = 1 and Var(X) = 5, (a) Find E[X^2]. (b)Derive the expression for Var(ax + b) where a and b is constant. c) Find Var(2x+4) and E[2x+4]. If E[X] = 1 and Var(X) = 5, (a) Find E[X^2]. (b)Derive the expression for Var(ax + b) where a and b is constant. c) Find Var(2x+4) and E[2x+4].

Explanation / Answer

Here' the answer to the question:

a. We now that

Var(X) = E(X^2) - E(X)^2

5 = E(X^2) - 1^2

E(X^2) = 6

b. Var(ax+b) = E((ax+b)^2) - E(ax+b)^2

= E(a^2x^2 + b^2 + 2abx) - (a^2*E(x)^2 + b^2+ 2abxE(x) )

= a^2 E(x^2)+ b^2 +2 abE(x) - a^2E(x)^2 - b^2 -2abxE(x)

= a^2 E(x^2) - a^2 E(X)^2

= a^2 (E (x^2) - E(X)^2)

= a^2 Var(x)

So, Var(ax + b) = a^2 Var(x)

<hence, proved>

c.Var(2x+4) = 2^2 Var(X) + var(4) = 4*Var(x) + 0 = 4*5 = 20

E[2x+4] = 2*E(X) + E(4) = 2*1+4 = 6

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