Expected value of a variable with piecewise pdf

Question

If I have that the pdf of a variable is: $$f_Y(y) = \begin{cases} \frac{1}{2}, & \text{if $y=0$} \\ \frac{1}{4a}, & \text{if $a<y<2a$} \\ \frac{1}{4a}, & \text{if 4a<y<5a}\\ 0, & OW \end{cases}$$ where $a>0$ constant, how can I find the Expected value of Y? In particular I am not sure how to handle the part where y=0.

Answer 1

You need to use sums for the parts where $Y$ is discrete, and integrate over parts where $Y$ is continuous. The formula you need to use here is $$\Bbb{E}[Y]=f_{Y}(0)\times 0 + \int_{a}^{2a} yf_{Y} (y)\, dy + \int_{4a}^{5a}yf_{Y}(y)\, dy.$$