Introduction

随机变量和概率之间一一对应的关系就是分布。

This chapter examines a set of distributions commonly applied to financial data and used by risk managers.

Two types of Common Univariate Random Variable Distributions

“Univariate” 是一个统计学术语，指的是只涉及单个变量的情况。一个 univariate 随机变量是指只涉及单个变量的随机变量，而不考虑其他变量的影响。

Discrete parametric distributions【离散参数分布】(PMF, E[.] and V[.])
- Bernoulli
- Binomial
- Poisson
Continuous parametric distributions(CDF, E[.] and V[.])
- Continuous uniform distribution【均匀分布】
- Normal distribution【正态分布】
- Lognormal distribution【对数正态分布】
- Student’s t-distribution【t分布】
- Chi-square distribution【卡方分布】
- F-distribution.【F分布】

Discrete random variables【重点】

Bernoulli Distribution

Bernoulli Distribution: It is a discrete distribution for random variables that produces one of two values: 0 or 1 .

抛一次硬币实验。

It applies to any problem with a binary outcome.
PMF
$$
f(X=1)=p \quad f(X=0)=1-p
$$
Expectation and variance

	Expectation	Variance
Bernoulli random variable	p	p(1-p)

MF Graph and CDF Graph for Bernoulli Distribution

The left panel shows the probability mass functions of two Bernoulli random variables, one with p = 0.5 and the other with p = 0.9. The right panel shows the cumulative distribution functions for the same two variables.

Binomial Distribution【二项分布/必考】

Binomial Distribution: The probability of $x$ successes in $n$ trails

独立重复n次二项分布成功的次数。

A binomial random variable measures the total number of successes from $n$ independent Bernoulli random variables
$PMF$
$$
f_X(x)=C_n^x p^x(1-p)^{n-x}=\frac{n !}{x !(n-x) !} p^x(1-p)^{n-x}
$$
Expectation and variance

	Expectation	Variance
Binomial random variable	np	np(1-p)

PMF Graph and CDF Graph for Bernoulli Distribution

The left panel shows the PMF of two binomial distributions.

The solid line is the PDF of a normal random variable, N(np, np(1- p))= N(10, 6), that approximates the PMF of the B(25, 0.4). The right panel shows the CDFs of the two binomials.

Poisson Distribution

当一件事发生的概率很小时，用二项分布来计算会产生很大的数值误差。

二项分布描述了在一系列独立的重复试验中成功事件发生的概率分布，比如投硬币时正面朝上的次数。然而，当试验次数非常大而成功概率非常小时，计算二项分布的概率变得非常复杂和耗时，因为需要计算大量的阶乘。

泊松分布作为对二项分布的近似，通过假设平均发生率相同但试验次数趋于无穷大的情况，简化了计算过程。泊松分布可以通过指数函数和阶乘的近似计算得到，避免了复杂的阶乘运算，并且在一定的条件下能够较好地逼近二项分布的概率分布。

因此，泊松分布可以看作是对于二项分布在特定条件下的近似，弥补了计算复杂性过高的缺陷，提供了一种简化计算的方法。然而，需要注意的是，泊松分布只适用于二项分布中试验次数非常大、成功概率非常小的情况，对于其他情况的近似效果可能不好。

Poisson Distribution: used to measure counts of events over fixed time spans. Such as the number of hurricanes in a century, or the number of phone calls in a day.

$$
f_X(X=k)=\frac{(\lambda)^k}{k !} e^{-\lambda}
$$
$\lambda$ indicates average number of the occurrence per interval, also called hazard rate

If $X_1 \sim Poisson(\lambda_1)$, and $X_2 \sim$ Poisson $(\lambda_2)$ are independent, and $Y=$ $X_1+X_2$, then $Y \sim$ Poisson $(\lambda_1+\lambda_2)$.

Expectation and variance

	Expectation	Variance
Poisson random variable	λ	λ

PMF Graph and CDF Graph for Poisson Distribution

The left panel shows the PMF for two values of a, 3 and 15. The right panel shows the corresponding CDFs.

Continuous random variables

Uniform Distribution

Uniform Distribution: A uniform distribution assumes that any value within the range $[a, b]$ is equally likely to occur.

For all $x_1<x_2$, we have:

$$
P(x_1 \leq X \leq x_2)=\frac{\min (x_2, b)-\max (x_1, a)}{b-a}
$$

Expectation and variance

	Expectation	Variance
Uniform random variable	(a +b)/2	$(b-a)^2 /12$

PDF Graph and CDF Graph for Uniform Distribution

The left panel shows the probability density functions of a standard uniform(i.e., U(0, 1)) and a uniform between 1/2 and 3. The right panel shows the cumulative distribution functions corresponding to these two random variables.

Normal

Normal (Gaussian, bell shaped) Distribution: the most commonly used distribution in risk management.

PDF(bell shaped)
$$
f(x)=\frac{1}{\sqrt{2 \pi} \sigma} \cdot e^{\frac{1}{2 \sigma^2}(x-\mu)^2}
$$
CDF: There is no closed form of normal distribution, fast numerical approximations are widely applied in practice.
Expectation and variance

	Expectation	Variance
Normal random variable	μ	$σ^2$

PDF Graph for Normal Distribution

● Symmetrical distribution: skewness = 0; kurtosis = 3.

● The tails get thin and go to zero but extend infinitely.

Normal distribution in practice

① Sums of independent normally distributed random variables are also normally distributed.

② Standardization and Z-table Application

③ Key quantiles in normal distribution

④ Approximating discrete random variables to normal random variable

(1) Sums of independent normally distributed random variables are also normally distributed

If $X \sim N(\mu_1, \sigma_1^2), {Y} \sim N(\mu_2, \sigma_2^2)$ and they are independent, then

$$
\mathrm{aX}+\mathrm{bY} \sim N\left(a \mu_1+b \mu_2, a^2 \sigma_1^2+b^2 \sigma_2^2\right)
$$

When applied to portfolio, $a$ and $b$ are usually asset weights

Example:A 50￡ million prudent fund (PF) is merged with a 200￡ million aggressive fund (AF). The return of ${PF} \sim {N}(0.03,0.07^2)$ and the return of $A F \sim N(0.07,0.15^2)$. Assuming the returns are independent, what is the distribution of the portfolio return?

Correct Answer: $R_p \sim N(0.062,0.1208^2)$

(2) Standardization and Z-table Application

Standardization: if $X \sim N(\mu, \sigma^2)$, then
$$
Z=\frac{X-\mu}{\sigma} \sim N(0,1)
$$
Check Z-table to find CDF
$$
\Phi\left(\frac{X-\mu}{\sigma}\right)
$$

Example:

If $X \sim N(70,9)$, compute the probability of $\mathrm{X} \leq 64.12$.
Solution:

$$
Z=(X-\mu) / \sigma=(64.12-70) / 3=-1.96
$$

$$
P(Z \leq-1.96)=0.0250
$$

$$
P(X \leq 64.12)=0.0250
$$

(3) Key quantiles in normal distribution

Approximately 68% of all observations fall in the interval $\mu \pm \sigma$
Approximately 90% of all observations fall in the interval $\mu \pm 1.65 \sigma$
Approximately 95% of all observations fall in the interval $\mu \pm 1.96 \sigma$
Approximately 99% of all observations fall in the interval $\mu \pm 2.58 \sigma$

(4) Approximating discrete random variables to normal random variable

A binomial random variable can approximate to a normal random variable, $N(n p, n p(1-p)$ if both

$$
\begin{aligned}
& n p \geq 10 \
& n(1-p) \geq 10 .
\end{aligned}
$$

The Poisson random variable can approximate to a normal random variable, N(λ,λ) if

λ is large, commonly applied when λ≥1000.

Lognormal Distribution

Lognormal Distribution: A variable $Y$ is said to be log-normally distributed if the natural logarithm of $Y$ is normally distributed.

An important property of the log-normal distribution is that it can never be negative. E.g.
The Black-Scholes Model assumes that the price of the underlying asset is lognormally distributed.
If InX is normal, then $X$ is lognormal; if a variable is lognormal, its natural log is normal, that is, $Y=\exp (X)$, where $X \sim N({\mu, \sigma^2})$ :
PDF:

$$
f_Y(y)=\frac{1}{y \sqrt{2 \pi \sigma^2}} \exp \left(-\frac{(\ln y-\mu)^2}{2 \sigma^2}\right)
$$

Expectation and variance

	Expectation	Variance
lognormal random variable	$e^{(μ+{\sigma}^2/2)}$	$(e^{\sigma^2}-1) {e}^{2 \mu+\sigma^2}$

DF Graph and CDF Graph for lognormal Distribution

● The left panel shows the PDFs of three log-normal random variables, two of which have μ = 8%,and two witho = 20%which are typical of annual equity returns. The right panel shows the corresponding CDFs.

Chi-Square

Chi-Square $\left(\chi^2\right)$ Distribution

Developed for testing hypotheses of positive parameters
If we have $k$ independent standard normal variables, $Z_1, Z_2, \ldots, Z_{k \prime}$ then the sum of their squares has a chi-squared distribution.
$$
\sum Z_i^2=Z_1^2+Z_2^2+\cdots+Z_k^2 \sim \chi_{(k)}^2
$$
Expectation and variance

	Expectation	Variance
Chi-Square random variable	k	2k

PDF Graph and CDF Graph for Chi-Square Distribution

● The left panel shows the PDFs of x2 distributed random variables with k=1,3 or 5 The right panel shows the corresponding CDFs.

t Distribution

t Distribution (Student’s t distribution): The Student’s t distribution is closely related to the normal, but it has heavier tails.

● Developed for testing hypotheses using small samples.

● If Z is a standard normal variable and U is a chi-square variable with k degrees of freedom, then the random variable X follows a t-distribution with k degrees of freedom.

Expectation,Variance(>=1),Kurtosis(>=3)

	Expectation	Variance	Kurtosis
Student’s t random variable	0	$\frac{k}{k-2}$	$\frac{3(k-2)}{k-4}$

PDF Graph and CDF Graph for Student’s t Distribution

The left panel shows the PDF of a generalized Student’s t with four degrees of freedom and the PDF of a standard normal. The right panel shows the corresponding CDFs.

F-Distribution

Developed for testing hypotheses
If $U_1$,and $U_2$ are two independent Chi-Squared distributions with $k_1$and $k_2$ degrees of freedom, respectively, then X:

follows an F-distribution with parameters $k_1$ and $k_2$.

Expectation and variance*

PDF Graph and CDF Graph for F Distribution

The left panel shows the PDF of three F-distributed random variables. The right panel shows the corresponding CDFs.

Mixture distribution

The distribution that results from a weighted average distribution of density functions is known as a mixture distribution. More generally, we can create a distribution:

PDF, PMF, CDF and non-central moment can be drawn with weighed average function.

Aplication of mixture distribution

● This class of distributions allows multiple simple distributions to be combined to produce distributions with empirically important features (e.g., skewness and heavy tails)

●E.g. Two normal random variable N(0.1) and N(-1,4) with mixing weights of 0.5 and 0.5, respectively, can create a skewed distribution.