SP-1 / Lesson 5
Todays extra does not only contain a few non-trivial problems, but also contains some problems about the theory that you may have not yet covered or do not have much experience with (like e.g conditional expectation). It is fine to google and try to remember those, this is partially the point of sharing such questions with you today.
Problem 1.
There are two random normal variables $X \sim N(0, \sigma_1)$, and $Y \sim N(0, \sigma_2)$. Define $U = X + Y$. Compute $\mathbb{E}[X \, | \, U]$ as well as $\mathbb{E}[Y \, | \, U]$.
Problem 2.
We draw a person at random from the street. Then we keep drawing people until we find someone taller than the first person. What is the expected number of draws we have to wait?
Problem 3.
Recall what is the moment generating function, and the exponential distribution, and Gamma distributions. Prove that the sum of exponential random variables is a Gamma distribution.