Expected value and standard deviation

Expected value — the mean of a random variable

• The expected value $E(X)$, also written $\mu$, is the long-run average value of the random variable over very many repetitions.

$$E(X) = \mu = \sum x \cdot P(X = x)$$

• Read it as a weighted average: each value is weighted by how likely it is.
• The expected value need not be a possible value. The expected number of heads in 3 tosses is 1.5, and you can never observe 1.5 heads. This is not an error — it is an average, not a prediction.

Calculating it

• Work in a column, and the arithmetic stays clean:

• Check the probability column sums to 1 before you trust the total.

Variance and standard deviation

• The variance measures spread — how far values typically fall from the mean. The version to use is:

$$\text{Var}(X) = \sigma^2 = \sum x^2 \cdot P(X = x) - \left[E(X)\right]^2$$

• The standard deviation is its square root, and it is in the same units as the variable, which is why it is the figure to quote:

$$\sigma = \sqrt{\text{Var}(X)}$$

• Method: build a third column of $x^2 P(X = x)$, total it, then subtract $\mu^2$.
• Sanity checks: the variance is never negative, and the standard deviation should be comparable to the typical distance of values from the mean. If $\sigma$ comes out larger than the whole range of the variable, you have made an error.

Interpreting the two together

• $\mu$ tells you where the distribution sits; $\sigma$ tells you how spread out it is.
• In context, always give both with units and say what they mean:
  • "On average the machine produces 2.4 rejects per hour, with a standard deviation of 1.3 rejects — so an hour with 4 rejects is unusual but not remarkable."

Scaling and shifting a random variable

• If a new variable is built from $X$ by multiplying and adding constants, $Y = aX + b$:

$$E(aX + b) = a\,E(X) + b$$

$$\text{Var}(aX + b) = a^2\,\text{Var}(X) \qquad \text{so} \qquad \text{sd}(aX+b) = |a| \times \text{sd}(X)$$

• Adding a constant shifts the mean but not the spread. Giving every customer $5 off raises nobody's variability.
• Multiplying scales both, but the variance scales by $a^2$ while the standard deviation scales by $a$ — because standard deviation is in the original units and variance is in squared units.

Adding independent random variables

• For independent $X$ and $Y$:

$$E(X + Y) = E(X) + E(Y)$$

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

• Means always add. Variances add only when the variables are independent — this is where a question tests whether you have thought about the context.
• Note the trap: the variance of $X + Y$ (two independent items) is not the same as the variance of $2X$ (one item doubled).

$$\text{Var}(X_1 + X_2) = 2\,\text{Var}(X) \qquad \text{but} \qquad \text{Var}(2X) = 4\,\text{Var}(X)$$

• Two separate randomly chosen items partly cancel each other's variation; doubling one item doubles its deviation as well.