Assuming all points in the triangle
are uniformly distributed, we have the joint density

where
is the area of the triangle
.
(a)

(where
is the portion of
for which
is between 0 and 5)



(b) Generalizing the previous result, we have

(this time with
being the portion of
where
for some
between 0 and 8)



That is, the CDF of
is

or

(c) Obtain the PDF by differentiating the CDF:


(d) Compute the expectation of
:
![E[X]=\displaystyle\int_0^8xf_X(x)\,\mathrm dx](https://img.qammunity.org/2021/formulas/mathematics/college/pwtzsbe81f6o7grr40rozx9ia08aei4z8n.png)
![E[X]=\displaystyle\frac1{32}\int_0^8x(8-x)\,\mathrm dx=\boxed{\frac83}](https://img.qammunity.org/2021/formulas/mathematics/college/qjlc5cuzhdxpcq657kdbgpzdrzb79j6kws.png)