5 Continuous Distributions

5.1 Uniform distributions

Here is the density and distribution function of the uniform distribution on \([0,1]\).

5.2 Normal distributions

Here is the density and distribution function of the normal distribution with mean 0 and variance 1 (i.e., the standard normal distribution).

5.3 Exponential distributions

Here is the density and distribution function of the exponential distribution with rate 1.

5.4 Normal approximation to the binomial

The De Moivre - Laplace theorem says that, as n increases while p remains fixed, the binomial distribution with parameters \((n, p)\) is well-approximated with the normal distribution with same mean (\(np\)) and same variance (\(n p (1-p)\)).

To corroborate this numerically, we fix \(p=0.10\) and vary \(n\) in \(\{10, 30, 100\}\). We can see that the approximation is already very good when \(n=100\).