Functions of Random Variables

The Expected Value Rule

If $g(X)$ is a function of random variable $X$:

$$E[g(X)] = \sum_x g(x) \cdot p_X(x)$$

This formula lets you find the expected value of $g(X)$ without first finding the PMF of $g(X)$.

Key Insight: Cannot Interchange Expectation and Functions

In general:

$$E[g(X)] \neq g(E[X])$$

You cannot reason on averages when dealing with non-linear functions.

Example: Reciprocal

If $X$ is a random variable, then:

$$E\left[\frac{1}{X}\right] \neq \frac{1}{E[X]}$$

From the travel example: $E[T] = E[200/V] = 100.5$, but $200/E[V] = 200/100.5 \approx 2$.

These are completely different!

Exceptions: Linear Functions

For linear functions $g(X) = aX + b$:

$$E[aX + b] = aE[X] + b$$

This is the linearity of expectation. It always works, even for dependent random variables.

Temperature Conversion

If $X$ is temperature in Celsius and $Y = 1.8X + 32$ is temperature in Fahrenheit:

$$E[Y] = 1.8E[X] + 32$$

You can first convert to Fahrenheit, then average. Or average in Celsius, then convert. Either way gives the same result.

Multiple Random Variables

For multiple random variables, linearity still holds:

$$E[X + Y + Z] = E[X] + E[Y] + E[Z]$$

SAT Example

If $X, Y, Z$ are scores on three SAT sections:

• Total score $= X + Y + Z$
• Average total $= E[X] + E[Y] + E[Z]$

You can average each section separately, then add. Or add scores for each student, then average. Same result.

Product of Independent Random Variables

If $X$ and $Y$ are independent:

$$E[XY] = E[X] \cdot E[Y]$$

Why Does This Work?

If $X$ and $Y$ are independent, knowing $X$ tells you nothing about $Y$. Therefore, knowing $X$ tells you nothing about $g(Y)$ for any function $g$.

So if $X$ and $Y$ are independent, then $g(X)$ and $h(Y)$ are also independent, and:

$$E[g(X)h(Y)] = E[g(X)] \cdot E[h(Y)]$$

Variance of Functions

Scaling

$$\text{Var}(aX) = a^2 \text{Var}(X)$$

The variance scales by the square of the scaling factor.

Shifting

$$\text{Var}(X + b) = \text{Var}(X)$$

Adding a constant shifts the distribution but doesn’t change its spread.

Sum of Independent RVs

If $X$ and $Y$ are independent:

$$\text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y)$$

Warning: This fails for dependent variables!

Example: Dependent Variables

If $X = Y$, then:

• $\text{Var}(X + Y) = \text{Var}(2X) = 4\text{Var}(X)$
• But $\text{Var}(X) + \text{Var}(Y) = 2\text{Var}(X)$

These are different!

Indicator Variables

An indicator variable takes values 0 or 1, indicating whether an event occurred.

Binomial Example

For $n$ coin flips with probability $p$ of heads:

Let $X_i = 1$ if flip $i$ is heads, 0 otherwise.

Then $X = X_1 + X_2 + \cdots + X_n$ counts total heads.

$$E[X] = E[X_1] + E[X_2] + \cdots + E[X_n] = np$$

This is much easier than summing over the binomial PMF!