Random Variables: formula sheet

Key formulas

PMF definition

p(x) = P(X = x)

Each p(x) ≥ 0 for a discrete variable.

PMF total probability

Σ p(x) = 1

Sum over every possible value of X. Use this to find a missing probability or constant.

Discrete probability of a set

P(a ≤ X ≤ b) = Σ p(x) for all x from a to b

Include or exclude endpoints exactly as the inequality says.

PDF non-negativity

f(x) ≥ 0

A density can exceed 1. It cannot be negative.

PDF total area

∫ f(x) dx = 1 over all x

Use this to solve for an unknown constant in f(x).

Continuous probability

P(a ≤ X ≤ b) = ∫ f(x) dx from a to b

Area under the curve. For a rectangle or triangle, use geometry instead of integrating.

Single point

P(X = a) = 0 for continuous X

So P(X ≤ a) = P(X < a).

Definition of the CDF

F(x) = P(X ≤ x)

Applies to discrete and continuous variables.

CDF from a PDF (continuous)

F(x) = ∫ f(t) dt, integrated from −∞ to x

For a uniform variable on [a, b], F(x) = (x − a) ÷ (b − a) for a ≤ x ≤ b.

PDF from a CDF

f(x) = dF(x) ÷ dx

Valid where F is differentiable.

CDF from a PMF (discrete)

F(x) = Σ P(X = xᵢ) for all xᵢ ≤ x

A step function that jumps at each possible value.

Interval probability

P(a < X ≤ b) = F(b) − F(a)

For a continuous variable, the endpoints do not matter.

Upper tail

P(X > x) = 1 − F(x)

Used for loss tails and exceedance probabilities.

Quantile (inverse CDF)

x_p = F⁻¹(p), where F(x_p) = p

For a continuous, strictly increasing F the quantile is unique.

Normal quantile

x_p = μ + σ × z_p

z_p = N⁻¹(p). Common values: z(0.95) = 1.645, z(0.975) = 1.96, z(0.99) = 2.326.

Properties of a CDF

0 ≤ F(x) ≤ 1; F(−∞) = 0; F(+∞) = 1; F non-decreasing

A function violating any of these is not a CDF.

Expected value (discrete)

E(X) = Σ xᵢ · P(xᵢ)

Probabilities must sum to 1. For a continuous variable the sum becomes an integral of x·f(x).

Variance (definition)

Var(X) = σ² = E[(X − μ)²] = Σ (xᵢ − μ)² · P(xᵢ)

μ = E(X). Always zero or positive.

Variance (shortcut)

Var(X) = E(X²) − [E(X)]²

Faster for most calculations. E(X²) = Σ xᵢ² · P(xᵢ). Do not confuse it with [E(X)]².

Standard deviation

σ = √Var(X)

Same units as X.

Linear transformation: mean

E(a + bX) = a + b·E(X)

a and b are constants.

Linear transformation: variance

Var(a + bX) = b² · Var(X)

The constant a drops out. Standard deviation of a + bX is |b| · σ.

Expectation of a sum

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

Holds whether or not X and Y are independent.

Expectation of a function

E[g(X)] = Σ g(xᵢ) · P(xᵢ)

In general E[g(X)] ≠ g(E[X]). For example, E(X²) ≠ [E(X)]².

Raw moment

k-th raw moment = E(Xᵏ)

First raw moment is the mean μ.

Central moment

k-th central moment = E[(X − μ)ᵏ]

First is 0; second is variance σ².

Variance from raw moments

σ² = E(X²) − [E(X)]²

Handy when raw moments are given.

Skewness

Skew = E[(X − μ)³] ÷ σ³

Unitless. 0 for symmetric distributions. Positive means right tail.

Kurtosis

Kurt = E[(X − μ)⁴] ÷ σ⁴

Normal distribution = 3.

Excess kurtosis

Excess kurtosis = Kurt − 3

Above 0 is leptokurtic (fat tails); below 0 is platykurtic.

Marginal probability

P(X = x) = Σy P(X = x, Y = y)

Sum the joint probabilities across the other variable. Row or column totals in a table.

Conditional probability

P(Y = y | X = x) = P(X = x, Y = y) ÷ P(X = x)

Defined only when P(X = x) > 0.

Independence

P(X = x, Y = y) = P(X = x) × P(Y = y) for all x, y

Must hold in every cell. Equivalent to P(Y | X) = P(Y).

Covariance

Cov(X,Y) = E[(X − μX)(Y − μY)] = E[XY] − E[X]E[Y]

Cov(X,X) = Var(X). Units are the product of the units of X and Y.

Correlation

ρ = Cov(X,Y) ÷ (σX × σY)

Always between -1 and +1. Measures linear dependence only.

Variance of a sum

Var(aX + bY) = a²Var(X) + b²Var(Y) + 2ab Cov(X,Y)

For a difference, the covariance term has the opposite sign when a = b = 1: Var(X − Y) = Var X + Var Y − 2Cov.

Covariance scaling

Cov(aX + b, cY + d) = ac Cov(X,Y)

Constants added do not change covariance. Correlation is unchanged by positive scaling.

Expected value of a sum

E(aX + bY) = aE(X) + bE(Y)

Holds whether or not X and Y are independent.

Bernoulli mean and variance

E(X) = p; Var(X) = p(1 − p)

X = 1 with probability p, X = 0 with probability 1 − p.

Binomial probability

P(X = k) = [n! ÷ (k!(n − k)!)] × p^k × (1 − p)^(n − k)

k = 0, 1, ..., n. Trials independent, same p.

Binomial mean and variance

E(X) = np; Var(X) = np(1 − p)

Standard deviation = √[np(1 − p)].

Poisson probability

P(X = k) = e^(−λ) × λ^k ÷ k!

k = 0, 1, 2, ... λ is the average number of events in the period.

Poisson mean and variance

E(X) = λ; Var(X) = λ

Scale λ to the period asked: a rate of 3 per month is 9 per quarter.

Continuous uniform on [a, b]

f(x) = 1 ÷ (b − a); E(X) = (a + b) ÷ 2; Var(X) = (b − a)² ÷ 12

P(c ≤ X ≤ d) = (d − c) ÷ (b − a) for a ≤ c ≤ d ≤ b.

Discrete uniform on integers 1 to n

E(X) = (n + 1) ÷ 2; Var(X) = (n² − 1) ÷ 12

Each value has probability 1 ÷ n.

Poisson approximation to binomial

λ = np

Reasonable when n is large and p is small.

Quick revision

• PMF gives P(X = x) for discrete variables; probabilities must be non-negative and sum to 1.
• PDF gives density, not probability; for a continuous variable P(X = x) = 0 and probability is the area under the curve.
• CDF: F(x) = P(X ≤ x); it never decreases and runs from 0 to 1.
• P(a < X ≤ b) = F(b) − F(a).
• The α-quantile is the value x where F(x) = α; VaR is a quantile of the loss distribution.
• E(aX + b) = aE(X) + b and Var(aX + b) = a²Var(X).
• Var(X) = E(X²) − [E(X)]²; standard deviation is √Var(X).
• Cov(X,Y) = E(XY) − E(X)E(Y); correlation lies between −1 and +1.
• Var(X + Y) = Var(X) + Var(Y) + 2Cov(X,Y).
• Zero correlation does not imply independence; independence does imply zero covariance (where it exists).
• Normal distribution: skewness 0, kurtosis 3; kurtosis above 3 means fatter tails.
• Binomial mean = np and variance = np(1 − p); Poisson mean and variance both equal λ.

Common mistakes

• Reading a PDF height as a probability. Fix: For continuous X, only area is probability. The height f(x) is a density and can exceed 1.
• Giving P(X = a) a positive value for a continuous variable. Fix: A single point has zero width and so zero area. Probability exists only over intervals.
• Treating the PDF value as a probability. Fix: For a continuous variable only areas under the PDF are probabilities. Use the CDF or integrate. P(X = x) = 0.
• Using F(x) when the question asks for P(X > x). Fix: Write the event first. P(X > x) = 1 − F(x).
• Writing Var(a + bX) = a + b·Var(X) or b·Var(X). Fix: Remember that variance is in squared units. The constant a vanishes and b is squared: Var(a + bX) = b²Var(X).
• Using [E(X)]² in place of E(X²). Fix: Square each outcome first, then weight by probability. The difference E(X²) − [E(X)]² is exactly the variance.
• Treating kurtosis of 3 as fat-tailed. Fix: Compare with 3, or use excess kurtosis and compare with 0.
• Dividing by variance instead of σ³ or σ⁴ powers of the standard deviation. Fix: Take the square root of variance first, then cube or raise to the fourth.
• Dividing by the wrong total for a conditional probability Fix: Divide by the marginal of the event you are conditioning on, the one after the "|" sign.
• Concluding independence from zero correlation Fix: Zero correlation only rules out linear dependence. Check the product rule in the cells; for example, Y = X² can have zero correlation with X yet be fully dependent.

Exam tips

• First decide discrete or continuous. Many wrong options are built from using the wrong rule.
• If an option says a continuous variable has a positive probability at one exact value, it is wrong.
• Use the sum-to-1 or area-to-1 rule to find missing constants before anything else.
• Use geometry for flat or triangular densities. It saves time over 100 questions in 4 hours.
• Read inequality signs closely on discrete questions: < versus ≤ changes the answer.
• Read whether the question gives a value and wants a probability, or gives a probability and wants a value. This decides CDF versus inverse CDF.
• Memorise z-values for 90%, 95%, 97.5% and 99%. They save time on every quantile question.
• Convert tail wording carefully: 'exceeded with 5% probability' and 'at the 95th percentile' describe the same upper quantile.