2022-08-24 E, Var Practice
Use the following table of probabilities. For the first part, assume that $X$ and $Y$ are chosen independently, so you pick one time randomly for $X$ and then one time randomly for $Y$.
Independent X and Y
| prob: | 0.35 | 0.20 | 0.20 | 0.25 |
|---|---|---|---|---|
| X | 10 | 8 | 0 | 4 |
| Y | 80 | -100 | 20 | 40 |
Questions
- $E[ X ]$
- Probability that $X=10$ and $Y\ge 0$.
- Probability that $X \cdot Y > 100$.
- Probability that $X>5$ given that $Y>0$. (See note.)
- $E[Y]$
- $E[X^2]$
- $E[Y^2]$
- $\Var(X)$
- $\Var(Y)$
Stretch
-
Suppose, unlike above, that both $X$ and $Y$ are chosen at the same time, so for example, you get $X=10$ and $Y=80$ exactly 35% of the time. Calculate $E[X\cdot Y]$ in this case.
-
Use the definition of $E[ X ]$ (see below) and basic math properties (name them!!) in order to prove that $$E[k \cdot X] = k \cdot E[ X ],$$ when $k$ is a constant.
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Find an example of random variables $X$ and $Y$ where $$E[ X \cdot Y ] \not= E[ X ] \cdot E[ Y ].$$ They shouldn’t be independent.
Official Definition of Expectation
This is an optional section.
In order to make an argument, you need to work with a specific definition of expectation.
Given a function $f(x)$, one way to define the expectation of $f(x)$ is $$ E\left[ f(x) \right] = \sum_x p(x) f(x) $$