1. For each of the following functions how do you determine whether it is a valid probabilty density function??????????????

(a) $\displaystyle f(x) = \frac{x}{2} , ~~~~~~~~ 0\leq x\leq 2$

(b) $\displaystyle f(x) = 4e^{-4x} , ~~~~~~~~~ x\geq 0$

(c) $\displaystyle f(x) = e^{x} + 1 , ~~~~~~~~~ 0\leq x \leq 1$

2.

(a) Find the value of a that makes $\displaystyle f(x) = ax(1-x)$ for $\displaystyle 0\leq x\leq 1$ a valid probability density function

(b) Evaluate the mean of this distribution

(c) Evaluate the variance of this distribution

Thanks for help