Question

Consider the following probability distribution for X. 7 0.1 0.1 0.2 003 0.2 0.1 (i) (ii) (iii) (iv) Find E(X). Find E(2X3+4x) Determine the MGF of X. Calculate Var (X) using MGF of X.

Answer 1

i)

E(X)=$\sum$xP(x)=-3*(0.1)+(-1)*0.2+1*0.3+3*0.2+5*0.1+7*0.1=1.6

ii)

E(2x³+4X) =$\sum$(2x³+4X)*P(x)=(2*(-3)³+4*(-3))*(0.1)+(2*(-1)³+4*(-1))*0.2+(2*(1)³+4*1)*0.3+(2*3³+4*3)*0.2+(2*5³+4*5)*0.1+(2*7³+4*2)*0.1=984

iii)

MGF =M_x(t)=E(e^tx) =$\sum$e^tx*P(x)=0.1*e^-3t+0.2*e^-t+0.3*e^t+0.2*e^3t+0.1*e^5t+0.1*e^7t

here first derivation of mgf =M_x'(t)=-0.3*e^-3t-0.2*e^-t+0.3*e^t+0.6*e^3t+0.5*e^5t+0.7*e^7t

therefore E(X)=M_x'(0)=-0.3*e^-3*0-0.2*e^-0+0.3*e⁰+0.6*e^3*0+0.5*e^5*0+0.7*e^7*0 = 1.6

second derivative of mgf M_x ''(t)=0.9*e^-3t+0.2*e^-t+0.3*e^t+1.8*e^3t+2.5*e^5t+4.9*e^7t

E(X²)=M_x ''(0)=0.9*e^-3*0+0.2*e^-0+0.3*e⁰+1.8*e^3*0+2.5*e^5*0+4.9*e^7*0 =10.6

hence Var(X)=E(X²)-(E(X))² =10.6-1.6² =8.04