Answer 2 If a and b arc constants and X a discrete random va

Answer

2. If a and b arc constants and X a discrete random variable, prove that E(aX + b) = aE(X) + b. (Hint: This is the discrete ease for Theorem 4.2 on page 116 of hour text book.]

Solution

if x1,x2,x3,x4..........,xn are discrete random variables, having mean = E(x)

E(x)= (x1+x2+x3+......+xn)/n...........................(1)

Now,

For,

y = ax+b

so,

y1 = ax1+b

y2 = ax2+b

y3= ax3+b

.....

......

yn = axn+b

So,

E(y) = (y1+y2+y3+...........+yn)/n

E(y) = (ax1+b+ax2+b+ax3+b+.............+axn+b)/n

E(y) = [a(x1+x2+x3+.....xn)+nb]/n

E(y) = a*[x1+x2+x3+....+xn]/n +b

E(y) = a*E(x) + b

Hence it is proved