Question

The probability density function of a random variable X is f

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]