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.
-
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
$$ E\left[ f(x) \right] = \sum_x p(x) f(x) $$