2.9 Computing Different Distributions.

Suppose that random variables \(X\) and \(Y\) are jointly continuous, with joint density function given by,

\[ f(x,y) = \begin{cases} c, & 0 \leq x \leq 1, 0 \leq y \leq x \\ 0, & otherwise \end{cases} \]

where \(c\) is a constant.

1. Draw a graph showing the region of the X-Y plane with positive probability density.
2. What is the constant \(c\)?
3. Compute the marginal density function for \(X\). (Be sure to write a complete expression)
4. Compute the conditional density function for \(Y\), conditional on \(X=x\). (Be sure to specify for what values of \(x\) this is defined)