Here is the cdf of an absolutely continuous random variable \(X\):

\[ F(x;\,\alpha,\,\theta) = \begin{cases} 1-\left(\frac{\theta}{x + \theta}\right)^\alpha & x >0 \\ 0 & \text{else}. \end{cases} \]

The parameters \(\alpha\) and \(\theta\) are just positive constants.

1. Find the pdf of \(X\) and plot it for \(\alpha=1,\, 2,\, 3\) and \(\theta = 1\);
2. Compute the median of \(X\);
3. Compute \(E(X)\). Is it finite for all values of the parameters?
4. Compute \(\textrm{var}(X)\). Is it finite for all values of the parameters?
5. Fix \((\alpha,\,\theta) = (3,\, 100)\) and compute \(P(X > 75\,|\, X > 50)\).

Tip

On Problem Set 5 you met some cute alternatives for computing expected values, and they may be helpful here.