Distributions: A Compendium (FRM Part 1)
1. Context
Here, we provide a quick summary and reference for various discrete (Binomial, Poisson) and continuous (Uniform, Exponential, Normal, Student’s t, Chi-Squared, F-Distribution) distributions in the FRM curriculum. The exam does not require you to memorize all formulas for pdf/cdf of a distribution, but it’s beneficial to have a basic understanding of what the parameter(s) mean, their impact, the mean and standard deviation and practical usage of each distribution. The details of the related reading are given below:
| Area | Quantitative Analysis |
| Reading | Distributions |
| Reference | Miller, Michael, Ch4. Distributions. In Mathematics and Statistics for Financial Risk Management, 2nd Edition, Hoboken, NJ: John Wiley & Sons, 2013. |
1. Bernoulli Distribution
| Parameters | $p$ (probability of success, probability of failure is $1-p$) |
| $$f(X=x;p)=p^x(1-p)^{1-x}$$ for $x=0,1$ | |
| Mean | $p$ |
| Variance | $p(1-p)$ |
| CDF | $$F(x)=\left\{ \begin{matrix} 1-p \mbox{, if } x=0 \\ 1 \mbox{, if } x=1 \\ \end{matrix} \right. $$ |
| Properties | A discrete distribution that is the building block of the Binomial distribution. |
| Usage | Appropriate to model events with two outcomes: a ‘success’ or a ‘failure’. Used for modeling default (success: default, failure: no-default) or to build binomial trees to numerically price derivatives and fixed income securities. |
2. Binomial Distribution
| Parameters | $p$ (probability of success, probability of failure is $1-p$) and $n$ (number of trials) |
| $$f(X=x;p)=\left( \begin{matrix} n \\ x \\ \end{matrix} \right) p^x(1-p)^{1-x}$$ for $x=0,1,2,…,n$ where $$\left( \begin{matrix} n \\ x \\ \end{matrix} \right) = \frac{n!}{(n-x)!x!}$$ | |
| Mean | $np$ |
| Variance | $np(1-p)$ |
| CDF | $$F(x\prime)=\sum_{x=0}^{x=x\prime}\left( \begin{matrix} n \\ x \\ \end{matrix} \right) p^x(1-p)^{1-x}$$ |
| Properties | A binomially distributed variable is constructed by chaining together $n$ independent trials, where each trial can be modeled as a Bernoulli variable. The name binomial actually comes from the fact that the PDF of distribution resembles the ($x+1$)’th term of a binomial expansion. |
| Usage | You will encounter binomial applications in applications such as backtesting to model number of exceedances and in modeling number of credit defaults in a portfolio context. |
3. Poisson Distribution
| Parameters | $\lambda$ (average or mean rate of occurrence of events) |
| $$f(X=x;\lambda)=\frac{\lambda^x e^{-\lambda}}{x!}$$ for $x=0,1,2,…$ | |
| Mean | $\lambda$ |
| Variance | $\lambda$ |
| CDF | $$F(x\prime)=\sum_{x=0}^{x=x\prime}\frac{\lambda^x e^{-\lambda}}{x!}$$ |
| Properties | Poisson distribution depends on only one parameter, which equals both it’s mean and variance. Poisson can be linked to and is a good approximation to the binomial distribution under the conditions $n\rightarrow \infty$, $p \rightarrow 0$ and $np \rightarrow \lambda$. It can also be linked to exponential distribution – if number of occurrence of events is modelled as a Poisson variable, the time between two events follows the exponential distribution. |
| Usage | Poisson distributed random variable is used to model the number of events occuring over a stated time interval. When used as an approximation for binomial distribution, it leads us to formulating reduced form models of default. |
4. Uniform Distribution
| Parameters | $a$ (lower limit) and $b$ (upper limit) |
| $$f(x;a,b)= \left\{ \begin{matrix} \frac{1}{b-a} \mbox{, if } a < x < b \\ 0 \mbox{, elsewhere} \\ \end{matrix} \right. $$ | |
| Mean | $$\frac{a+b}{2}$$ |
| Variance | $$\frac{1}{12}(a-b)^2$$ |
| CDF | $$F(x)=\frac{x-a}{b-a}$$ |
| Properties | Very simple continuous distribution. |
| Usage | Uniform distribution is used in copulas to model the grade of any random variable X, defined as $U=F(X)$, where $F()$ denotes the cumulative distribution function of $X$. |
5. Normal Distribution
| Parameters | $\mu$ (location) and $\sigma$ (scale) |
| $$f(x;\mu,\sigma)=\frac{1}{\sigma\sqrt{2\pi}}\exp \left[ -\frac{(x-\mu)^2}{2\sigma^2} \right]$$ | |
| Mean | $\mu$ |
| Variance | $\sigma^2$ |
| CDF | Use numerical approximations, NORM.DIST() in Excel |
| Properties | Symmetric (skewness = 0) distribution with mean = mode = median. Excess kurtosis of this distribution is 0, and is not suitable for uses where fat tails are to be modelled. It’s standard form $Z \sim N(0,1)$ is more commonly used wherein every normally distributed variable $X \sim N(\mu,\sigma)$ is first standardized using: $Z=\frac{X-\mu}{\sigma}$. A few properties to remember are: $$\begin{matrix} X\sim N(\mu_X, \sigma_X), Y=aX+b \\ \Rightarrow Y\sim N\left(a\mu_X +b,a\sigma_X\right) \\ X\sim N(\mu_X, \sigma_X), Y\sim N(\mu_Y, \sigma_Y) \\ \Rightarrow X+Y \sim N\left(\mu_X + \mu_Y, \sqrt{\sigma^2_X+\sigma^2_Y}\right) \\ \end{matrix}$$ (if $X$ and $Y$ are uncorrelated) |
| Usage | Also called Gaussian distribution or bell curve, normal distribution is most popular distribution in finance used in applications owing to it’s simplicity, symmetricity and parsimony. Applications range from modelling returns and in risk management for modelling profit-and-loss distributions to compute risk measures like VaR. Owing to Central Limit Theorem, normal distribution also finds lot of use in econometrics (hypothesis testing of large sample regression coefficients). |