A bicycle manufacturing company makes a particular type of bike. Each child bike requires 4 hours to build and 4 hours to test.
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I'm going to assume the joint density function is
a. In order for to be a proper probability density function, the integral over its support must be 1.
b. You get the marginal density by integrating the joint density over all possible values of
:
c. We have
d. We have
and by definition of conditional probability,
e. We can find the expectation of using the marginal distribution found earlier.
f. This part is cut off, but if you're supposed to find the expectation of , there are several ways to do so.
- Compute the marginal density of
, then directly compute the expected value.
- Compute the conditional density of
, then use the law of total expectation.
The law of total expectation says
We have
This random variable is undefined only when which is outside the support of
, so we have