Question

Which of the following statements are true for all random variables X, Y and fixed numbers a, b ER? lí Z If E [aX]=ElbX), then a = b.

The answer can be one of many

What is selected was marked incorrect.

Answer 1

E[X + a] = E[X + b]

=> E[X] + a = E[X] + b

=> a = b

If E[X + a] = E[X + b], then a = b is correct

E[aX] = E[bX]

=> aE[X] = bE[X]

=> a = b if E[X] $\ne$ 0

If E[aX] = E[bX], then a = b is correct only when E[X] $\ne$ 0

E[X]  $\ne$ E[Y]

=> aE[X]  $\ne$ aE[Y]

=> aE[X] + b $\ne$ aE[Y] + b

E[aX + b] $\ne$ E[aY + b]

If E[X]  $\ne$ E[Y] then E[aX + b] $\ne$ E[aY + b] is correct.

Thus the second option, If E[aX] = E[bX], then a = b seems to be incorrect.

It is not valid when E[X] = 0