# Calculating expected value and variance of a probability density function

If I was given a probability density function:

$$f(y) = \left\{\begin{array}{ll}\frac{3y^2(4-y)}{64} & \textrm{for } 0 \leq y \leq 4\\ 0 & \textrm{elsewhere} \end{array}\right.$$

for expected value would that just be the following integral? $$\int_{0}^{4} yf(y)\,\textrm{d}y$$

I do not know how I would calculate the variance though. Any tips?

Thanks

• Yes, that is fine for the expected value. Then find the expected value of $Y^2$ in similar fashion. Use the two expectations to get the variance. – soakley Mar 10 '14 at 16:52
• @soakley would I have to change the y every time? so I would that formula 4 times because I have 1,2,3,4? – user125627 Mar 10 '14 at 16:54
• @user125627 You are evaluating a definite integral, so the probability distribution is continuous, not discrete (i.e. values can be 1.5, 2.78 etc) – user130512 Mar 10 '14 at 16:59
• @user130512 I am getting a little confusing because shouldn't the expected value be the sum of all the values, but wouldn't that mean I will have to change y multiple times? – user125627 Mar 10 '14 at 17:01
• The summation formula is only used for discrete random variables. You need to use integration. Look at the given answer. – soakley Mar 10 '14 at 17:02

For the expected value, you need to evaluate the integral $$\int_0^4 yf(y) dy =\int_0^4 {3y^3(4-y) \over 64} dy$$
You can either find the variance directly by applying the law of the unconscious statistician with $g(y)=(y-{\rm E}[Y])^2$, that is, $$\mathrm{Var}(Y)={\rm E}[(Y-{\rm E}[Y])^2]=\int_0^4g(y)f(y)\,\mathrm dy,$$ or you could find ${\rm E}[Y^2]$ by the same formula with $g(y)=y^2$ and then use that $$\mathrm{Var}(Y)={\rm E}[Y^2]-{\rm E}[Y]^2.$$