## Probability

### The Analysis of Data, volume 1

Random Variables: Exercises

## 2.7. Exercises

1. Prove that $\P'$ is a probability function.
2. Give a concrete example of a random variable that is neither discrete nor continuous.
3. Prove Corollary 2.1.1.
4. Consider a continuous RV $X$ whose density is $f_X(x)=x$ for $x\in(0,1)$ and 0 otherwise. Compute its expectation and variance.
5. Consider an RV $Y=X^2$ where $X$ is defined in (3). Compute the pdf of $Y$ and its expectation using both formulas in Proposition 2.3.1.
6. We define the support size of a discrete RV $X$ as the number of values it can achieve with nonzero probability (that number may be infinity). What is the relationship between the support sizes of a discrete RV $X$ and of $g(X)$?