Generating Continuous Random Variables¶

To generate a random variable \(X\), We utilise it's cumulative distribution function \(F\). Note that \(F\) is strictly increasing from \(0\) to \(1\) and the probability density \(f\) and \(F\) have a one to one mapping, i.e., \(F\) has an inverse \(F^{-1}\). So, to get \(X\), we first generate \(U \in (0, 1)\) and then \(X = F^{-1}(U)\).

Exponential Random Variable¶

The distribution function \(F = 1 - exp(-\lambda x)\). Then, the inverse is

\[ \begin{aligned} x &= -\frac{1}{\lambda}log(1 - F)\newline \text{or, }\quad X &= \frac{1}{\lambda}log(1 - U) \end{aligned} \]

and \(X\) will follow the exponential distribution. Further, replacing \(U\) with \(1-U\) will still remain a uniform distribution, meaning \(X\) will still be exponential.

Normal Random Variable¶

Consider \(X, Y \sim \mathcal{N}(0, 1)\). Then,

\[ \begin{aligned} f_{XY}(x,y) = \frac{1}{2\pi}exp(-\frac{x^{2}+y^{2}}{2}) \end{aligned} \]

is the joint distribution. \((X, Y)\) is a point on the cartesian plane and thus, will have an equivalent polar coordinate \((R, \Theta)\) implying \(R^{2} = X^{2} + Y^{2}\) which is a \(\chi_{2}^{2}\) variable. From relation between gamma and \(\chi_{2}^{2}\) variable,

\[ \begin{aligned} f_{R^{2}}(r) &= \frac{1}{2}exp(-\frac{r}{2})\newline \text{and} \quad f_{XY}(x,y) &= \frac{1}{2\pi} exp(-\frac{r}{2}) \quad \text{when} \quad x^{2} + y^{2} = r \end{aligned} \]

Now, \(R\) and \(\Theta\) are independent variables and can be used to generate \(X\) and \(Y\). \(\Theta\) is uniformly distributed in \([0, 2\pi]\). Taking the inverse of cumulative distribution of \(R^{2}\) (Note that \(f_{R^{2}}(r)\) is a distribution on \(R^{2}\) and not \(R\)),

\[ \begin{aligned} R^{2} &= -2log(1 - U_{1})\newline \Theta &= 2\pi U_{2} \end{aligned} \]

where \(U_{1}\) and \(U_{2}\) are uniform random variables. Using the transformation back to cartesian coordinates from polar ones,

\[ \begin{aligned} X &= \sqrt{-2log(1 - U_{1})}cos(2\pi U_{2})\newline Y &= \sqrt{-2log(1 - U_{1})}sin(2\pi U_{2}) \end{aligned} \]

This approach is called Box-Muller method. To generate standard normals with mean \(\mu\) and variance \(\sigma^{2}\), we simply return \(\mu + \sigma X, \mu+\sigma Y\).

Normal Random Variable from Uniform Distribution¶

Using the central limit theorem, we can add up multiple uniform distributions to get a normal random variable.

\[ \begin{aligned} Y = \bigg(\sum_{i=1}^{12}X_{i} \bigg) - 6 \sim \mathcal{N}(0,1) \end{aligned} \]

because the mean of 12 uniform random variables is \(12 * 0.5 = 6\) and the variance of sum of 12 independent random variables is \(12 \times (1-0)^{2}/12 = 1\). The same equation with 30 uniform distributions will become

\[ \begin{aligned} Y = \sqrt{\frac{2}{5}}\bigg[\bigg(\sum_{i=1}^{30}X_{i} \bigg) - 15\bigg] \sim \mathcal{N}(0,1) \end{aligned} \]

and combining more variables will increase the accuracy of the approximation, but also increase computational time.