Random Variables

In practice, you never need to manipulate $\mathcal{F}$ explicitly in interview problems. What matters is the probability measure $P$ and its properties below.

Boole's inequality is the go-to tool to upper bound the probability that at least one of many bad events occurs. If each $P(A_i)$ is small, the union is also small.

This theorem justifies passing limits through probability signs for monotone sequences — a technique used often in convergence proofs.

In practice, a random variable is simply a quantity whose value depends on the outcome of a random experiment. The measurability condition ensures that $P(X \leq x)$ is always well-defined.

The CDF uniquely characterizes the distribution of $X$. Two random variables with the same CDF have the same distribution.

Pairwise independence does not imply mutual independence. A classic counterexample: toss two fair coins, let $A = \{\text{first is H}\}$, $B = \{\text{second is H}\}$, $C = \{\text{both same}\}$ — pairwise independent but not mutually independent.

The tail formula $E[X] = \int_0^\infty P(X > t)\,dt$ is extremely powerful for non-negative integer-valued RVs: $$. Use it for geometric distributions, waiting times, and coupon collector-type problems.

In interviews, always reach for $\text{Var}(X) = E[X^2] - (E[X])^2$ to compute variance — it avoids expanding $$ directly. Compute via the transfert formula.

For a normal distribution: $\gamma_1 = 0$ (symmetric) and $\gamma_2 = 0$ (excess kurtosis = 0). A distribution with ${\gamma }_2>0$ is (heavy tails) — relevant in finance where asset returns exhibit excess kurtosis.

Markov only requires $X \geq 0$ and knowledge of $E[X]$ — it is the weakest but most general bound. In interviews, use it when you only know the mean and need an upper bound on a tail probability.

Chebyshev is the standard tool to prove the Law of Large Numbers and to bound probabilities when only mean and variance are known. It is distribution-free — valid for any $X$ with finite variance.

Cantelli is strictly stronger than Chebyshev on one side: it gives an upper bound on the one-sided tail $P(X \geq \mu + \lambda)$ without the factor of 2. Useful when the direction of deviation matters.

Jensen is the key inequality when dealing with convex/concave transformations of expectations — log-returns, utility functions, option pricing bounds. Whenever you see $E[f(X)]$ vs $f(E[X])$, ask yourself if $f$ is convex or concave.

Cauchy-Schwarz is the special case $p = q = 2$. Hölder generalizes to any conjugate pair. The case $p = 1, q = \infty$ gives $E[\mathrm{\mid }XY\mathrm{\mid }]\le E[\mathrm{\mid }X\mathrm{\mid }]\cdot \mathrm{\parallel }$.

PGFs are most useful for sums of independent discrete RVs and branching processes. The factorization property $G_{X+Y} = G_X \cdot G_Y$ is the key tool.

The MGF does not always exist (e.g. heavy-tailed distributions like Cauchy). In that case, use the characteristic function instead.

In interviews, the MGF is used to: (1) identify a distribution by matching its MGF to a known one, (2) compute moments quickly via differentiation, (3) prove that a sum of independents follows a known law.

The characteristic function is the Fourier transform of the density. It always exists, making it more general than the MGF. The inversion formula recovers the density: $f_X(x) = \dfrac{1}{2\pi}\int_{-\infty}^{+\infty} e^{-itx}\varphi_X(t)\,dt$.

The Laplace transform is essentially the MGF evaluated at $-s$: $\mathcal{L}_X(s) = M_X(-s)$. It is preferred for non-negative RVs (exponential, gamma) and in queuing theory / reliability contexts.