TheprobabilitydensityfunctionofarandomvariableXisf(X)(a)showthatE(aX+b)=aE(X)+b(b)Var(aX+b)=a^2Var(X)(c)Var(X)=E(X^2)-{E(X)}^2

Question

The probability density function of a random variable X is f(X)
(a)show that E(aX+b)=aE(X)+b
(b)Var(aX+b)=a^2Var(X)
(c)Var(X)=E(X^2)-{E(X)}^2

Answer 1

（a):Proof:E[aX+b] = Sum pi(axi +b)
= Sum (pi (axi) + pi (b))
= Sum (pi axi)+ Sum ( pi b)
= aSum (pi xi)+ bSum ( pi),Sum( pi) = 1,所以 Sumaxi = aSxi
= aE[X] + b
(b) Proof:
Var(X) = E([X-E(X)]2
= E(X^2)-2XE(X) + E(X)^2,x=E(x)
= E(X^2) -2E(X)E(X) + E(X)^2
= E(X^2)-E(X)^2
(c)
Proof:Var[aX+b]
= E([(aX + b)- E(aX + b)]^2)
= (a^2E[X^2] + 2abE[X] +b^2) – (a^2E[X]^2+2abE[X]+b^2)
= a^2E[X^2] - a^2E[X]^2
= a^2(E[X^2] - E[X]^2)
= a^2Var[X]