Probability & Statistics Notes

Last reviewed 16 Sept 2026 · 9 min read

Descriptive statistics

Measure	Definition	Notes
Arithmetic mean	$\overset{x}{ˉ} = \frac{\sum x _{i}}{n}$ ; grouped: $\frac{\sum f _{i} x _{i}}{\sum f _{i}}$	Affected by extreme values
Median	Middle value of ordered data; grouped: $L + \frac{N /2 - C}{f} h$	Robust to outliers
Mode	Most frequent value; grouped: $L + \frac{f _{1} - f _{0}}{2 f _{1} - f _{0} - f _{2}} h$	May not be unique
Geometric mean	$(x_{1} x_{2} \dots x_{n})^{1/ n}$	Rates, ratios
Harmonic mean	$\frac{n}{\sum 1/ x _{i}}$	Average speeds

Empirical relation (moderately skewed data): Mode ≈ 3 Median − 2 Mean.

Range = maximum − minimum.
Mean deviation $= \frac{\sum ∣ x _{i} - x ˉ ∣}{n}$ (minimum when taken about the median).
Variance $σ^{2} = \frac{\sum ( x _{i} - x ˉ ) ^{2}}{n}$ (population); sample variance $s^{2} = \frac{\sum ( x _{i} - x ˉ ) ^{2}}{n - 1}$ .
Standard deviation $σ = variance$ .
Coefficient of variation $C V = \frac{σ}{x ˉ} \times 100%$ — relative variability (used to compare consistency; e.g. concrete quality control).
Adding a constant to all values does not change SD; multiplying by $k$ multiplies SD by $∣ k ∣$ .

Skewness — asymmetry; positive (right tail longer): mean > median > mode; negative: mean < median < mode. Karl Pearson coefficient $= \frac{mean - mode}{σ}$ .
Kurtosis — peakedness; normal distribution has $β_{2} = 3$ (mesokurtic); > 3 leptokurtic; < 3 platykurtic.

Random experiment, sample space $S$ , event $A \subseteq S$ .
Classical probability $P (A) = \frac{n ( A )}{n ( S )}$ for equally likely outcomes.
Axioms: $0 \leq P (A) \leq 1$ ; $P (S) = 1$ ; for mutually exclusive events $P (A \cup B) = P (A) + P (B)$ .
$P (\overset{ˉ}{A}) = 1 - P (A)$ .

FormulaProbability rules

Addition rule: $P (A \cup B) = P (A) + P (B) - P (A \cap B)$

Conditional probability: $P (A ∣ B) = \frac{P ( A \cap B )}{P ( B )}$

Multiplication rule: $P (A \cap B) = P (A) P (B ∣ A)$

Independent events: $P (A \cap B) = P (A) P (B)$

Total probability: $P (B) = \sum_{i} P (A_{i}) P (B ∣ A_{i})$ for a partition $A_{1}, A_{2}, \dots$

Bayes' theorem:

P (A_{i} ∣ B) = \frac{P ( A _{i} ) P ( B ∣ A _{i} )}{\sum _{j} P ( A _{j} ) P ( B ∣ A _{j} )}

Mutually exclusive events cannot occur together ( $P (A \cap B) = 0$ ); independent events do not affect each other — non-trivial events cannot be both.

Type	Description	Mean	Variance
Discrete	Probability mass function $p (x)$ , $\sum p (x) = 1$	$E (X) = \sum x p (x)$	$E (X^{2}) - [E (X)]^{2}$
Continuous	Probability density function $f (x) \geq 0$ , $\int f (x) d x = 1$ ; $P (a < X < b) = \int_{a}^{b} f d x$	$\int x f (x) d x$	$\int x^{2} f d x - μ^{2}$

Cumulative distribution function $F (x) = P (X \leq x)$ ; $f (x) = F^{'} (x)$ .
$E (a X + b) = a E (X) + b$ ; $Var (a X + b) = a^{2} Var (X)$ .
For independent $X, Y$ : $Var (X \pm Y) = Var (X) + Var (Y)$ .

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