Probability Distributions

Probability distributions provide a mathematical framework for modeling the uncertainty inherent in financial markets. They are essential for tasks ranging from asset pricing and risk management to portfolio optimization and algorithmic trading.

I. Foundational Concepts

A Random Variable (RV) is a variable whose value is a numerical outcome of a random phenomenon. RVs are classified as Discrete (countable outcomes, e.g., number of defaults) or Continuous (uncountable outcomes over a range, e.g., asset price).

Concept	Discrete RV	Continuous RV	Description
Probability Function	Probability Mass Function (PMF), $f(x)$	Probability Density Function (PDF), $f(x)$	Defines the probability of a discrete outcome or the relative likelihood of a continuous outcome.
Cumulative Function	Cumulative Distribution Function (CDF), $F(x)$	Cumulative Distribution Function (CDF), $F(x)$	Gives the probability that the RV takes a value less than or equal to $x$ : $F(x) = P(X \le x)$ .
Expected Value	$\mathbb{E}[X] = \sum x_i f(x_i)$	$\mathbb{E}[X] = \int x f(x) dx$	The weighted average of all possible values, representing the long-run average.
Variance	$\text{Var}(X) = \mathbb{E}[(X - \mu)^2]$	$\text{Var}(X) = \mathbb{E}[(X - \mu)^2]$	Measures the dispersion or spread of the distribution around the mean ( $\mu$ ).

Moment Generating Functions (MGF)

The Moment Generating Function (MGF), $M_X(\theta) = \mathbb{E}[e^{\theta X}]$ , is a powerful tool.

Utility: The $k$ -th moment of the distribution ( $\mathbb{E}[X^k]$ ) can be found by taking the $k$ -th derivative of the MGF and evaluating it at $\theta=0$ .
Sum of RVs: The MGF of the sum of independent random variables is the product of their individual MGFs: $M_{X+Y}(\theta) = M_X(\theta) M_Y(\theta)$ .

II. Key Distributions in Statistics

The following table summarizes the most critical distributions, their parameters, and their relevance in financial modeling.

Name	Type	Application	PMF/PDF	$\mu$	$\sigma^2$
Bernoulli	Discrete	Modeling a single event outcome (e.g., default/no default, success/failure of a trade).	$f(t;p) = p^t (1-p)^{1-t}$	$p$	$p(1-p)$
Binomial	Discrete	Number of successes in a fixed number of trials (e.g., number of up-moves in a Binomial Option Pricing Model, credit risk modeling).	$f(t;n,p) = \binom{n}{t} p^t (1-p)^{n-t}$	$np$	$np(1-p)$
Poisson	Discrete	Modeling the number of rare events over a fixed time (e.g., number of trades, defaults, or jumps in a jump-diffusion model).	$f(t;\lambda) = \frac{\lambda^t e^{-\lambda}}{t!}$	$\lambda$	$\lambda$
Exponential	Continuous	Modeling the time until the next event in a Poisson process (e.g., time until default or time between trades).	$f(t;\lambda) = \lambda e^{-\lambda t} \mathbf{1}_{t \ge 0}$	$\frac{1}{\lambda}$	$\frac{1}{\lambda^2}$
Uniform	Continuous	Modeling uncertainty when all outcomes are equally likely (e.g., random number generation, simple Monte Carlo simulations).	$f(t;a,b) = \frac{1}{b-a} \mathbf{1}_{t \in [a,b]}$	$\frac{a+b}{2}$	$\frac{(b-a)^2}{12}$
Normal	Continuous	The distribution for modeling asset returns (log-returns) due to the CLT. Used in Markowitz portfolio theory and basic risk models.	$f(t) = \frac{1}{\sigma\sqrt{2\pi}} \exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)$	$\mu$	$\sigma^2$
Lognormal	Continuous	The distribution for modeling asset prices in the Black-Scholes-Merton model, as prices cannot be negative. If $X \sim N(\mu, \sigma^2)$ , then $Y = e^X \sim \text{Lognormal}$ .	$f(y) = \frac{1}{y\sigma\sqrt{2\pi}} \exp\left(-\frac{(\ln y-\mu)^2}{2\sigma^2}\right)$	$e^{\mu + \sigma^2/2}$	$e^{2\mu + \sigma^2}(e^{\sigma^2}-1)$
Student's t	Continuous	Used to model financial returns with heavy tails (fat tails), capturing extreme events more accurately than the Normal distribution. Parameter $\nu$ (degrees of freedom) controls tail thickness.	$f(t;\nu) \propto \left(1 + \frac{t^2}{\nu}\right)^{-\frac{\nu+1}{2}}$	0 (for $\nu>1$ )	$\frac{\nu}{\nu-2}$ (for $\nu>2$ )

Probability Distributions

I. Foundational Concepts

Moment Generating Functions (MGF)

II. Key Distributions in Statistics

Probability & Statistics