8. Let X be a random variable that takes on values between 0 and

c. That is, P[0≤ X ≤ c] = 1. Show that

$$Var(X) ≤ \frac{c^2}{4}$$

HINT: One approach is to first argue that

$$E[X^2] ≤ cE[X]$$

Then use this to show that

$$Var(X) = c^2[α(1-α)]$$ where α =

E[X]

/ c