# 6. Expected Value, Variance, and Standard Deviation of a Random Variable

## l. calculate and interpret the expected value, variance, and standard deviation of a random variable and of returns on a portfolio;

What is the definition of "expected value" and what is the mathematical equation?

- Expected value is the probability weighted average of the possible outcomes. More probable outcomes will have more weight.

- In finance, expected value is used a lot. E.g. "expected value of earnings per share", "dividend per share", "rate of return", etc

- $$E(X) = P(x_1)x_1 + P(x_2)x_2 + ... + P(x_n)x_n$$

What is the variance of an expected value (probability weighted average)?

- The variance of a random variable is the expected value (the probability-weighted average) of squared deviations from the random variable's expected value.

- $$\sigma^2(X) = E([X - E(X)]^2)$$

What does a variance = 0 mean?

Variance of zero means that the outcome is certain. Your scatterplot points all fall directly on the regression line, given the data you could exactly predict the outcome every time.

What does a variance > 0 mean?

Variance greater than zero means that there is dispersion in the outcomes. The higher the number the more dispersion.

What are variance and standard deviation measures of?

They are both measures of dispersion in the possible outcomes of the expected value of a random variable.

What is the math formula for total probability rule for expected value? (Think of the concert example)

- $$E(X) = E(X | S_1)P(S_1) + E(X | S_2)P(S_2) + ... + E(X | S_n)P(S_n)$$

- `X`

= revenue

- `S_1`

= favorable weather

- `S_2`

= moderate weather

- `S_3`

= bad weather

- Reads: The expected outcome of X equals the expected outcome of X given that scenario `S_1`

happens, times the probability of scenario `S_1`

happening, plus ... (until `S_n`

)

##### Source:

- CFA