Lecture 4. G) Normal
Normal Distribution
Recall: Random variable [math]X[/math] follows a normal distribution [math]N\left(\mu,\sigma^{2}\right)[/math] if it is continuous with pdf [math]f_{X}\left(\left.x\right|\mu,\sigma^{2}\right)=\frac{1}{\sqrt{2\pi\sigma^{2}}}\exp\left(-\frac{\left(x-\mu\right)^{2}}{2\sigma^{2}}\right),x\in\mathbb{R}[/math]
The normal distribution is by far the most important continuous distribution. Its main claim to fame is that it can be shown (as we will, later) that the average of a large number of random variables, under some conditions, are normally distributed. This result is called the central limit theorem.
Note that if [math]X\sim N\left(\mu,\sigma^{2}\right)[/math], [math]X=\mu+\sigma\widetilde{X}[/math] where [math]\widetilde{X}\sim N\left(0,1\right)[/math].
CDF
The cdf of the normal distribution does not admit a closed-form representation. However, we do use a short hand that relies on the [math]N\left(0,1\right)[/math] distribution:
[math]F_{X}\left(x\right)=P\left(X\leq x\right)=P\left(\mu+\sigma\widetilde{X}\leq x\right)=P\left(\widetilde{X}\leq\frac{x-\mu}{\sigma}\right)=\Phi\left(\frac{x-\mu}{\sigma}\right),[/math] where [math]\Phi\left(\cdot\right)[/math] is the standard normal cdf., i.e.,
[math]\Phi\left(x\right)=\int_{-\infty}^{x}\frac{1}{\sqrt{2\pi}}\exp\left(-\frac{t^{2}}{2}\right)dt,x\in\mathbb{R}[/math].
