Advanced probability

What the problems look like

A few statements from this skill. Answers, hints and worked solutions live in the practice session.

1. Warm-up

   A random variable $X$ has density $f(x) = \frac{2x}{25}$ on $[0, 5]$ and $f(x) = 0$ elsewhere. Compute the expectation $\mathbb{E}[X] = \int_0^{5} x\,f(x)\,dx$.

2. Warm-up

   A random variable $X$ has density $f(x) = \frac{2x}{16}$ on $[0, 4]$ and $f(x) = 0$ elsewhere. Compute the expectation $\mathbb{E}[X] = \int_0^{4} x\,f(x)\,dx$.

3. Warm-up

   A random variable $X$ has density $f(x) = \frac{2x}{49}$ on $[0, 7]$ and $f(x) = 0$ elsewhere. Compute the expectation $\mathbb{E}[X] = \int_0^{7} x\,f(x)\,dx$.

Problems shown under their original terms: Generated (template, solver-computed).