Expected Values: Moments, Mode, Median, and Percentiles | Free Exam P Lesson

Expected value gives average future lifetime. Median tells you what half the group exceeds. Mode gives the most common claim size. These are three different numbers from the same distribution, confusing them changes the reserve by millions.

Discrete:

$E [X] = x \sum x \cdot p_{X} (x)$

Continuous:

$E [X] = \int_{- \infty}^{\infty} x f_{X} (x) d x$

Linearity: $E [a X + b] = a E [X] + b$ .

HIGH-FREQUENCY: LOTUS is used in nearly every Exam P expected value calculation involving transformations.

$E [g (X)] = \int_{- \infty}^{\infty} g (x) f_{X} (x) d x (continuous)$

You do not need the distribution of $g (X)$ . Stay in "X-land."

The $k$ -th raw moment: $μ_{k}^{'} = E [X^{k}]$ .

The $k$ -th central moment: $μ_{k} = E [(X - μ)^{k}]$ .

The variance shortcut:

$Var (X) = E [X^{2}] - (E [X])^{2}$

$M_{X} (t) = E [e^{tX}]$

Moments from differentiation:

$E [X^{k}] = M_{X}^{(k)} (0)$

The mode maximizes $f_{X} (x)$ (continuous) or $p_{X} (x)$ (discrete). Set $f^{'} (x) = 0$ , confirm maximum. Always check endpoints on bounded domains.

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Common mistakes

Using $E [X^{2}] = (E [X])^{2}$ . For $N (3, 16)$ : $(E [X])^{2} = 9$ , but $E [X^{2}] = 25$ . Setting them equal gives $Var = 0$ . Trap: reporting zero variance.
Confusing median with mean. For $f (x) = 3 x^{2}$ on $(0, 1)$ : mean = 0.75, median = 0.794. For right-skewed (exponential, Pareto): mean > median. Trap: reporting 0.75 as the median.
Wrong direction for exponential percentile. The 90th percentile: $- θ ln (0.1)$ . Using $- θ ln (0.9)$ gives the 10th percentile. For $θ = 100$ : trap is 10.54 instead of 230.26.

Bottom line

LOTUS: $E [g (X)]$ uses the distribution of $X$ , not $g (X)$ , so stay in "X-land"
Expected value: sum or integrate $x$ against the PMF/PDF, and it is linear: $E [a X + b] = a E [X] + b$
Variance shortcut: $Var (X) = E [X^{2}] - (E [X])^{2}$ , compute each piece separately
MGF: $M_{X} (t) = E [e^{tX}]$ , with moments via $E [X^{k}] = M_{X}^{(k)} (0)$

Exam shortcut

When a problem gives the MGF, identify the distribution by matching known forms (normal, Poisson, exponential, binomial). Once identified, write down moments and percentiles directly. Differentiation is the fallback. "LOTUS: stay in X-land." Compute $E [g (X)]$ by integrating $g (x) f_{X} (x)$ . You never need the distribution of $g (X)$ . "Second minus squared." $Var = E [X^{2}] - (E [X])^{2}$ . Write this at the top of every variance calculation.

The full lesson (about 3,088 words, 21 min read) adds 4 worked examples, all 8 common mistakes, a self-check, free in the app.

Learning objectives

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