Distributions and Random Numbers¶

Feng Li

School of Statistics and Mathematics

Central University of Finance and Economics

Basic concepts of random numbers¶

Pseudo random numbers
- an algorithm for generating a sequence of numbers that approximates the properties of random numbers.
- The sequence is not truly random in that it is completely determined by a relatively small set of initial values, called the PRNG’s state.
- Pseudo random numbers are important in practice for their speed in number generation and their reproducibility.

Random seed

A random seed (or seed state, or just seed) is a number (or vector) used to initialize a pseudo random number generator.
The most important random numbers are from uniform distributed numbers.

> runif(n,a,b)
Numbers selected from a non-uniform probability distribution can be generated using a uniform distribution PRNG and a function that relates the two distributions.
Assume you have uniformly distributed random numbers from [0, 1], how do you extend it to [a, b]?

The normal density function $$f(x, \mu, \sigma) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2} }$$

> dnorm(x,mu,sigma)

> dnorm(x,mu,sigma, log=TRUE)
- In theory, dnorm(x,mu,sigma, log=TRUE)==log(dnorm(x,mu,sigma))
- but dnorm(x,mu,sigma, log=TRUE) but is more stable for very large values. Why?
- We love logs.

In [6]:

dnorm(100, mean=0, sd=1)
log(dnorm(100, mean=0, sd=1))

dnorm(100, mean=0, sd=1, log=TRUE)

-Inf

-5000.9189385332

The CDF (Cumulative Distribution Function) $$\Phi(x)\;=\int_{-\infty}^x f(t, \mu, \sigma) d t$$

> pnorm(q,mu,sigma)
The quantile (Given CDF, what is x?), i.e. $\Phi^{-1}(p)$

> qnorm(p,mu,sigma)
Random numbers from normal distribution

> rnorm(n,mu,sigma)

In [5]:

z = rnorm (100,mean=0, sd=1)
hist(z)

In [3]:

x  = seq(-5, 5, 0.1)
y = dnorm(x = x, mean = 0, sd = 1)
plot(x, y, col = "blue", type = "l", lwd = 4)

In [4]:

y1 = pnorm(q = x, mean = 0, sd = 1)
plot(x, y1, col = "blue", type = "l", lwd = 4)