05. Expected Values
Definition¶
Let \(X\) be a random variable. Then, \(E[X]=\sum_kk\cdot P_X(k)\)
This is a weighted average of possible values of X, where the weight of each value is the possibility of X taking that value.
Alternate form
You may also see \(E[X]\) expressed as \(\mu\). They are the same thing.
Roll of a single die
Let X= roll of a single die
\(E[X]=1\cdot\frac16+2\cdot\frac16+3\cdot\frac16+4\cdot\frac16+5\cdot\frac16+6\cdot\frac16=\frac72=3.5\)
Tirgul 4 Question 5
A pilot wants to insure his private plane worth 200,000 NIS. The insurance company estimates the probability of damage to the plane each year as follows:
- 10% chance of a 50k NIS loss
- 1% chance of a 100k NIS loss
- 0.2% chance of a 200k NIS loss
If the insurance company wants an expected annual profit of 500 NIS from insuring this plane, what annual premium should it charge the pilot?
Define \(p=\)annual premium. We wish to solve for this variable.
Define \(X=\) annual profit to company from pilot.
We make our table as follows, assigning \(k\) to be possible values of \(X\):
| \(k\) | \(P_X(k)\) |
|---|---|
| \(p\) | \(100\%-10\%-1\%-0.2\% = 88.8\%\) |
| \(p-50,000\) | 0.1 |
| \(p-100,000\) | 0.01 |
| \(p-200,000\) | 0.02 |
\(500=E[X]=(0.888)\cdot p+(0.1)(p-50,000)+(0.01)(p-100,000)+(0.002)(p-200,000)\)
—Numerous examples omitted—
How to Solve Expected Value Questions
Assuming \(X\) is the Random Variable
- Create a two column table, where the first column is labled \(k\), and the second, \(P_X(k)\).
- Write each possible value for \(X\) in the \(k\) column, and the probability of \(X\) taking that value in the \(P_X(k)\) column.
- Multiply each \(k\) by it’s corresponding \(P_X(k)\).
- Sum up the products
The value obtained in step 4 is the Expected Value.
Properties of Expected Values¶
- \(E[cX]=cE[X]\)
- \(E[X+c]=E[X]+c\)
- \(E[X+Y]=E[X]+E[Y]\)
- If \(X,Y\) are independent, then \(E[X\cdot Y]=E[X]\cdot E[Y]\)
Linearity of Expectation¶
The first three properties are called linearity of expectation.
They can be expressed as: