Random Variables & Expected Value

What is a random variable?

A random variable $X$ assigns a number to each outcome of a random process. Discrete random variables take countable values (a dice roll, a count of successes); continuous ones take any value in a range (a height, a waiting time).

Probability distribution

For a discrete random variable, the distribution lists each value with its probability. Every probability is between 0 and 1, and they sum to 1.

Example. $X$ = the number on a fair die. $P(X = k) = \frac{1}{6}$ for $k = 1, 2, \ldots, 6$.

Expected value (the mean of a random variable)

$E[X] = \mu = \sum_i x_i\, P(x_i)$

It's the long-run average if you repeated the process forever — each value weighted by its probability.

Example (fair die). $E[X] = (1 + 2 + 3 + 4 + 5 + 6)\cdot\frac{1}{6} = \frac{21}{6} = 3.5$.

Example (a game). Win $5 with probability $0.2$, otherwise lose $2:

$E[\text{profit}] = 0.2(5) + 0.8(-2) = 1 - 1.6 = -0.6$

You lose about 60 cents per play on average.

Variance and standard deviation

Variance measures spread around the mean:

$\mathrm{Var}(X) = \sigma^2 = \sum_i (x_i - \mu)^2\, P(x_i)$

and the standard deviation is $\sigma = \sqrt{\mathrm{Var}(X)}$.

Example. $X$ pays $0$ with probability $0.5$ and $10$ with probability $0.5$. Then $\mu = 5$, and

$\mathrm{Var}(X) = (0-5)^2(0.5) + (10-5)^2(0.5) = 12.5 + 12.5 = 25$

so $\sigma = 5$.

Transformation rules

For constants $a$ and $b$:

$E[aX + b] = a\,E[X] + b$ $\mathrm{Var}(aX + b) = a^2\,\mathrm{Var}(X)$

Adding a constant shifts the mean but leaves the spread alone; multiplying scales the mean by $a$ and the variance by $a^2$.

Sums of random variables

$E[X + Y] = E[X] + E[Y] \quad (\text{always})$ $\mathrm{Var}(X + Y) = \mathrm{Var}(X) + \mathrm{Var}(Y) \quad (\text{if independent})$

Means always add. Variances add only when the variables are independent.

Key takeaways

• A random variable turns outcomes into numbers, with a probability distribution that sums to 1.
• $E[X] = \sum x_i\,P(x_i)$ is the long-run average.
• $\mathrm{Var}(X) = \sum (x_i - \mu)^2 P(x_i)$; $\sigma$ is its square root.
• $E[aX + b] = aE[X] + b$ and $\mathrm{Var}(aX + b) = a^2\mathrm{Var}(X)$.
• Means always add; variances add for independent variables.