10.4. Normal Distribution

The normal density function is given by

\[f_X(x) = \frac{1}{\sqrt{2\pi} \sigma} \exp\left ( - \frac{(x - \mu)^2}{2\sigma^2} \right)\]

Probability Density Function

Evaluating the density function for different values of \(x\), \(\mu\) and \(\sigma\):

> dnorm(x=0, mean = 0, sd = 1)
[1] 0.3989423
> dnorm(x=-4:4, mean = 0, sd = 1)
[1] 0.0001338302 0.0044318484 0.0539909665 0.2419707245 0.3989422804 0.2419707245 0.0539909665 0.0044318484
[9] 0.0001338302
> dnorm(x=-3:5, mean = 1, sd = 1)
[1] 0.0001338302 0.0044318484 0.0539909665 0.2419707245 0.3989422804 0.2419707245 0.0539909665 0.0044318484
[9] 0.0001338302

We can use dnorm to plot the PDF of normal distribution:

> x <- seq(-4,4,by=0.01)
> y <- dnorm(x)
> plot(x,y, 'l', main ='Normal Distribution')

Cumulative Distribution Function

The function pnorm() is used to compute the CDF of normal distribution up to any point on the real line:

> pnorm(0)
[1] 0.5
> pnorm(1)
[1] 0.8413447
> pnorm(-1)
[1] 0.1586553
> pnorm(1, mean=1)
[1] 0.5
> pnorm(-4:4)
[1] 3.167124e-05 1.349898e-03 2.275013e-02 1.586553e-01 5.000000e-01 8.413447e-01 9.772499e-01 9.986501e-01
[9] 9.999683e-01

By default pnorm gives the integral of the PDF from :math:-\infty to q. It is also possible to compute the integral from q to \(\infty\) using the lower.tail parameter:

> pnorm(0, lower.tail = FALSE)
[1] 0.5
> pnorm(1, lower.tail = FALSE)
[1] 0.1586553
> pnorm(-1, lower.tail = FALSE)
[1] 0.8413447

Note that pnorm(x) + pnorm(x, lower.tail=FALSE)=1.

Quantile or Inverse Cumulative Distribution Function

We can use the qnorm function to compute the z-score for a given quantile value:

> qnorm(c(0, .25, .5, .75, 1))
[1]       -Inf -0.6744898  0.0000000  0.6744898        Inf
> qnorm(.5, mean=1)
[1] 1
> qnorm(pnorm(-3:3))
[1] -3 -2 -1  0  1  2  3

Finally, we use rnorm for generating random numbers from the normal distribution.

Hazard function

Hazard function is given by \(H(x) = - log (1 - F(x))\).

This can be computed as follows:

> q =1
> -log(pnorm(q, lower.tail = FALSE))
[1] 1.841022

Log likelihood

Log likelihood function is given by \(log (f(x))\).

This can be computed by:

> dnorm(x, log=TRUE)
[1] -0.9189385

10.4.1. Bivariate Normal Distribution

In this section, we will look at different ways to generate samples from bivariate normal distribution.

Let our random variable be denoted as X = (X1, X2). Let the number of samples to be generated be N.

The simplest case is when both X1 and X2 are independent standard normal variables:

> N <- 1000
> set.seed(123)
> samples <- matrix(rnorm(N*2), ncol=2)
> colMeans(samples)
[1] 0.01612787 0.04246525
> cov(samples)
          [,1]      [,2]
[1,] 0.9834589 0.0865909
[2,] 0.0865909 1.0194419

The next case is when the two variables are independent but have different means:

> mu1 <- 1
> mu2 <- 2
> samples <- cbind(rnorm(N, mean=mu1), rnorm(N, mean=mu2))
> colMeans(samples)
[1] 0.9798875 1.9908396
> cov(samples)
           [,1]       [,2]
[1,] 0.95718335 0.04908825
[2,] 0.04908825 0.98476186

There is a function called mvrnorm in the MASS package which is very flexible:

> mu1 <- 1
> mu2 <- 2
> mu <- c(mu1, mu2)
> sd1 <- 2
> sd2 <- 4
> corr <- 0.6
> Sigma <- matrix(c(sd1, corr, corr, sd2), nrow=2)
> library(MASS)
> N <- 10000
> samples <- mvrnorm(N, mu=mu, Sigma=Sigma)
> colMeans(samples)
[1] 0.9976949 2.0208528
> cov(samples)
          [,1]      [,2]
[1,] 1.9889508 0.6005303
[2,] 0.6005303 4.0516402