Continuous Distributions

Overview

Continuous distributions model random variables that take values in an uncountable set, characterized by a density function rather than a PMF. This topic covers the foundational tools (density, CDF, change of variable) and the full catalog of classical continuous distributions organized by family — uniform/exponential, normal, gamma, and heavy-tailed. Each distribution is presented with its key formulas, MGF, and typical interview use cases.

1. General Continuous Random Variable

1.1 Density & CDF

Definition

A random variable $X$ is continuous if there exists a non-negative function $f_X : \mathbb{R} \to \mathbb{R}_+$ , called the probability density function (PDF), such that for all $a \leq b$ :

\boxed{P(a \leq X \leq b) = \int_a^b f_X(x)\, dx}

Properties

$f_X(x) \geq 0$ for all $x$
$\displaystyle\int_{-\infty}^{+\infty} f_X(x)\, dx = 1$

Definition— CDF

\boxed{F_X(x) = P(X \leq x) = \int_{-\infty}^{x} f_X(t)\, dt}

Properties

$F_X$ is continuous, non-decreasing
$\lim_{x\to-\infty} F_X(x) = 0$ ,

1.2 Expectation & Transfert Formula

Formula— Expectation

\boxed{E[X] = \int_{-\infty}^{+\infty} x\, f_X(x)\, dx}

Formula— Transfert Formula

For any measurable function $g$ :

\boxed{E[g(X)] = \int_{-\infty}^{+\infty} g(x)\, f_X(x)\, dx}

Formula— Tail Formula (for

X \geq 0

)

\boxed{E[X] = \int_0^{+\infty} P(X > t)\, dt = \int_0^{+\infty} (1 - F_X(t))\, dt}

💡Tip

The tail formula avoids direct integration when $P(X > t)$ has a clean closed form — particularly useful for Exponential and other distributions where the survival function $1 - F_X(t)$ is simpler than the density.

1.3 Change of Variable

Theorem— Change of Variable (Univariate)

Let $X$ be a continuous RV with density $f_X$ and $Y = \varphi(X)$ where $\varphi$ is and differentiable. Then has density:

Theorem— Change of Variable (Multivariate)

Let $(X_1, \ldots, X_n)$ have joint density $f_X$ and where is a diffeomorphism. Then:

Remarque

For the non-monotone case (e.g. $Y = X^2$ ), split the domain into monotone pieces and sum the contributions: $f_Y(y) = \sum_i f_X(x_i) \cdot |dy/dx|^{-1}_{x=x_i}$ where are the preimages of .

Example

Let $X \sim \mathcal{N}(0,1)$ and $Y = e^X$ . Then $\varphi^{-1}(y) = \log y$ and , so:

2. Uniform & Exponential Family

2.1 Uniform $(a, b)$

Definition

$X$ is uniformly distributed on $[a, b]$ if it is equally likely to take any value in the interval:

\boxed{X \sim \mathcal{U}(a,b) \iff f_X(x) = \frac{1}{b-a}\, \mathbf{1}_{[a,b]}(x)}

Properties

$E[X] = \dfrac{a+b}{2}$
$\text{Var}(X) = \dfrac{(b-a)^2}{12}$

💡Tip

The uniform distribution is the foundation of simulation: if $U \sim \mathcal{U}(0,1)$ then $F_X^{-1}(U) \sim X$ for any distribution with invertible CDF (inverse transform method).

2.2 Exponential $(\lambda)$

Definition

The Exponential distribution models the waiting time between events in a Poisson process:

\boxed{X \sim \text{Exp}(\lambda) \iff f_X(x) = \lambda e^{-\lambda x}\, \mathbf{1}_{x \geq 0}}

Properties

$E[X] = \dfrac{1}{\lambda}$
$\text{Var}(X) = \dfrac{1}{\lambda^2}$

Theorem— Memoryless Property

\boxed{P(X > s + t \mid X > s) = P(X > t) \quad \forall s, t \geq 0}

Remarque

The memoryless property means past waiting time gives no information about future waiting time. It is the continuous analog of the Geometric distribution's memoryless property.

💡Tip

Use Exponential for: inter-arrival times in a Poisson process, time to failure, service times. If $X_1, \ldots, X_n$ are i.i.d. Exp $(\lambda)$ , then $\min (_{}$ — a very useful result for competition/race problems.

3. Normal Family

3.1 Normal $(\mu, \sigma^2)$

Definition

$X$ follows a Normal (Gaussian) distribution with mean $\mu$ and variance $\sigma^2 > 0$ :

X \sim N (μ, σ^{2}) ⟺ f_{X} (x)

Properties

$E[X] = \mu$
$\text{Var}(X) = \sigma^2$
$F_{X} (x) = Φ ⁣ (\frac{x}{}$

Remarque

For $X \sim \mathcal{N}(\mu, \sigma^2)$ : $P(\mu - \sigma \leq X \leq \mu + \sigma) \approx 68\%$ , , — the .

💡Tip

Key results to know instantly: $E[e^X] = e^{\mu + \sigma^2/2}$ for $X \sim N (μ, σ^{2})$ (from MGF at ). This is the foundation of the Black-Scholes formula.

3.2 Log-Normal $(\mu, \sigma^2)$

Definition

$X$ is Log-Normal with parameters $\mu \in \mathbb{R}$ and $\sigma^2 > 0$ if $\log X \sim \mathcal{N}(\mu, \sigma^2)$ :

Properties

$E[X] = e^{\mu + \sigma^2/2}$
$\text{Var}(X) = \left(e^{\sigma^2}-1\right)e^{2\mu+\sigma^2}$

Remarque

The Log-Normal has no MGF for $t > 0$ — a classic interview trap. Use the moment formula $E[X^k] = e^{k\mu + k^2\sigma^2/2}$ directly instead.

💡Tip

Log-Normal is the standard model for asset prices in finance: if log-returns are normal (Black-Scholes assumption), prices are log-normal. $E[S_T] = S_0 e^{\mu T + \sigma^2 T/2}$ is the expected stock price under this model.

4. Gamma Family

4.1 Gamma $(\alpha, \beta)$

Definition

The Gamma distribution generalizes the Exponential to the sum of $\alpha$ exponential waiting times. With shape $\alpha > 0$ and rate $\beta > 0$ :

X \sim Γ (α, β) ⟺ f_{X} (x) = \frac{β^{α}}{}

Properties

$E[X] = \dfrac{\alpha}{\beta}$
$\text{Var}(X) = \dfrac{\alpha}{\beta^2}$

💡Tip

Sum of $n$ i.i.d. Exp $(\lambda)$ variables $\sim \Gamma(n, \lambda)$ — use this to find the distribution of total waiting time for $n$ events in a Poisson process.

4.2 Chi-Squared $(\nu)$

Definition

The Chi-Squared distribution with $\nu$ degrees of freedom is the distribution of the sum of squares of $\nu$ independent standard normal variables:

\boxed{X \sim \chi^2(\nu) \iff X = Z_1^2 + Z_2^2 + \cdots + Z_\nu^2, \quad Z_i \overset{iid}{\sim} \mathcal{N}(0,1)}

Properties

$\chi^2(\nu) = \Gamma\!\left(\dfrac{\nu}{2}, \dfrac{1}{2}\right)$ (special case of Gamma)

💡Tip

$\chi^2(\nu)$ appears whenever you sum squared normals — sample variance, goodness-of-fit tests, confidence intervals for variance. For large $\nu$ : $\chi^2(\nu) \approx \mathcal{N}(\nu, 2\nu)$ by CLT.

4.3 Beta $(\alpha, \beta)$

Definition

The Beta distribution models a random variable on $[0,1]$ , with shape parameters $\alpha, \beta > 0$ :

X \sim Beta (α, β) ⟺ f_{X} (x) = \frac{x^{α - 1} (1 - x)^{β - 1}}{B}

Properties

$E[X] = \dfrac{\alpha}{\alpha+\beta}$
$\text{Var}(X) = \dfrac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}$

💡Tip

Use Beta for modeling probabilities or proportions (values in $[0,1]$ ). It is the conjugate prior for the Binomial in Bayesian inference — a key result in quantitative interviews involving Bayesian updating.

5. Heavy-Tailed & Derived Distributions

5.1 Student- $t(\nu)$

Definition

The Student- $t$ distribution with $\nu > 0$ degrees of freedom arises as the ratio of a standard normal to the square root of a scaled chi-squared:

\boxed{T = \frac{Z}{\sqrt{V/\nu}}, \quad Z \sim \mathcal{N}(0,1),\ V \sim \chi^2(\nu),\ Z \perp V}

Properties

$E[T] = 0$ for $\nu > 1$ , undefined for $\nu = 1$
$\text{Var}(T) = \dfrac{\nu}{\nu-2}$ for

💡Tip

Student- $t$ is used in statistics when the population variance is unknown. In interviews: $t(\nu)$ with small $\nu$ has heavy tails — important for modeling financial returns where extreme events are more frequent than the Normal predicts.

5.2 Fisher $F(d_1, d_2)$

Definition

The Fisher F-distribution with degrees of freedom $d_1, d_2 > 0$ is the ratio of two independent scaled chi-squared variables:

F = \frac{V_{1} / d_{1}}{V_{2} / d_{2}},

Properties

$E[F] = \dfrac{d_2}{d_2-2}$ for

💡Tip

The F-distribution appears in ANOVA and regression — testing whether multiple group means are equal, or whether a set of regression coefficients are jointly zero. Know it structurally: ratio of two chi-squared divided by their degrees of freedom.

5.3 Cauchy $(x_0, \gamma)$

Definition

The Cauchy distribution with location $x_0$ and scale $\gamma > 0$ :

\boxed{f_X(x) = \frac{1}{\pi\gamma\left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]}}

Properties

No mean, no variance — all moments of order $\geq 1$ are undefined
$F_X(x) = \dfrac{1}{\pi}\arctan\!\left(\dfrac{x-x_0}{\gamma}\right) + \dfrac{1}{2}$

Remarque

The Cauchy distribution is the canonical example of a distribution with undefined mean and variance. The sample mean of i.i.d. Cauchy variables does not converge — the Law of Large Numbers fails. This is a classic interview trap: "does the LLN apply?" → check whether $E[|X|] < \infty$ .

6. Quick Reference

Full Summary Table

Distribution	Parameters	$f_X(x)$	$E[X]$	$\text{Var}(X)$	$M_X(t)$

Key Relationships

Relationship	Statement
Exp → Gamma	Sum of $n$ i.i.d. Exp $(\lambda)$ $\sim \Gamma(n, \lambda)$
Normal → Chi-Squared	$\sum_{i=1}^\nu Z_i^2 \sim \chi^2(\nu)$ ,

Uniform $(a,b)$	$a < b$	$\frac{1}{b-a}$ on $[a,b]$	$\frac{a+b}{2}$	$\frac{(b-a)^2}{12}$	$\frac{e^{tb}-e^{ta}}{t(b-a)}$	Equal likelihood, simulation
Exponential $(\lambda)$	$\lambda > 0$	$\lambda e^{-\lambda x}$ , $x\geq 0$	$\frac{}{}$
Normal $(\mu,\sigma^2)$	$\mu\in\mathbb{R}$ , $\sigma^2>0$	$\frac{1}{}$
Log-Normal $(\mu,\sigma^2)$	$\mu\in\mathbb{R}$ , $\sigma^2>0$	$\frac{1}{}$
Gamma $(\alpha,\beta)$	$\alpha,\beta>0$	$\frac{\beta^\alpha}{\Gamma(\alpha)}x^{\alpha-1}e^{-\beta x}$
Chi-Squared $(\nu)$	$\nu>0$	$\Gamma(\nu/2, 1/2)$ density	$\nu$	$2\nu$
Beta $(\alpha,\beta)$	$\alpha,\beta>0$	$\frac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)}$ on
Student- $t(\nu)$	$\nu>0$	see above	$0$ ( $\nu>1$ )	$\frac{\nu}{\nu-2}$ ()
Fisher $F(d_1,d_2)$	$d_1,d_2>0$
Cauchy $(x_0,\gamma)$	$x_0\in\mathbb{R}$ , $\gamma>0$

Continuous Distributions

1. General Continuous Random Variable

1.1 Density & CDF

1.2 Expectation & Transfert Formula

1.3 Change of Variable

2. Uniform & Exponential Family

2.1 Uniform(a,b)(a, b)(a,b)

2.2 Exponential(λ)(\lambda)(λ)

3. Normal Family

3.1 Normal(μ,σ2)(\mu, \sigma^2)(μ,σ2)

3.2 Log-Normal(μ,σ2)(\mu, \sigma^2)(μ,σ2)

4. Gamma Family

4.1 Gamma(α,β)(\alpha, \beta)(α,β)

4.2 Chi-Squared(ν)(\nu)(ν)

4.3 Beta(α,β)(\alpha, \beta)(α,β)

5. Heavy-Tailed & Derived Distributions

5.1 Student-t(ν)t(\nu)t(ν)

5.2 Fisher F(d1,d2)F(d_1, d_2)F(d1​,d2​)

5.3 Cauchy(x0,γ)(x_0, \gamma)(x0​,γ)

6. Quick Reference

Full Summary Table

Key Relationships

2.1 Uniform $(a, b)$

2.2 Exponential $(\lambda)$

3.1 Normal $(\mu, \sigma^2)$

3.2 Log-Normal $(\mu, \sigma^2)$

4.1 Gamma $(\alpha, \beta)$

4.2 Chi-Squared $(\nu)$

4.3 Beta $(\alpha, \beta)$

5.1 Student- $t(\nu)$

5.2 Fisher $F(d_1, d_2)$

5.3 Cauchy $(x_0, \gamma)$