Feng Li

School of Statistics and Mathematics

Central University of Finance and Economics

**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)
```

0

-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)
```

Distribution Function in R

Student t:

`{p,d,q}t`

Chi squared:

`{p,d,q}chi`

Gamma:

`{p,d,q}gamma`

Exponential

`{p,d,q}exp`

- For a significance test, what distribution do you use?

- Jones (2009):
**Chapter 14, 15, 16**