Chapter 16 - Random Variables
| Tuesday, 29 November 2022 | |
| 2-minute read | |
| 302 words | |
Discrete vs Continuous
A discrete random variable has a countable number of outcomes. A continuous random variable can occur in infinitely many ways.
A continuous variable can fall anywhere within an interval, but a discrete variable must fall into a specific place.
Expected Value
An expected value of a variable can be based on a probability distribution.
Here is an example with life insurance.
| Outcome | Payout | Probability |
|---|---|---|
| Death | $10000 | 1/1000 |
| Disability | $5000 | 2/1000 |
| Neither | $0 | 997/1000 |
\[ E(x) = 1E4(\frac{1}{1E3}) + 5E3(\frac{2}{2E3}) + 0(\frac{997}{1E3}) = 20 \]
This means you expect to pay $20 per policy.
Formally, expected value can be written in the notation:
\[ \mu = E(x) = \Sigma [x \cdot p(x)] \]
Standard Deviation of a Random Variable
| Outcome | Payout | Probability | Deviation x - mu |
|---|---|---|---|
| Death | $10000 | 1/1000 | 10000 - 20 = 9800 |
| Disability | $5000 | 2/1000 | 5000 - 20 = 4980 |
| Neither | $0 | 997/1000 | 0 - 20 = -20 |
\[ \Sigma [(\textrm{Dev})^2 \cdot p(x)] \]
Example
A person who is 21+ is playing a card game with one standard deck of playing cards at a casino. The dealer will turn over one card at random. How much should the casino charge to play the game? What is the standard deviation of this dataset?
Here is the payout structure:
| Card | Payout | Probability | Deviation |
|---|---|---|---|
| Any red card | $5 | 1/2 | 5 - 9.23 = -4.23 |
| Black ace | $25 | 1/26 | 25 - 9.23 = 15.77 |
| Two of clubs | $300 | 1/52 | 300 - 9.23 = 290.77 |
| None | $0 | 0.557692308 | -9.23 |
\[ E(x) = \frac{1}{2}(5) + \frac{1}{26}(25) + \frac{1}{52}300 = 9.231 \]
Therefore, the casino should charge $9.24 or more to play the game to make a profit over time.
\[ \sigma = \frac{1}{2} (4.23)^2 + \frac{1}{26} (15.77)^2 + \frac{1}{52} = (290.77)^2 + 0.557692308(9.23)^2 = 41.13 \]
The standard deviation of this game is 41.13