Examples of distributions

Standard normal distribution#

There is no closed form expression for \(F(x)\), but the density function is

\[f(x) = (2\pi)^{-1/2}\exp(-x^2/2).\]

If \(X\) has the standard normal distribution, then \(EX =0\) and \(\text{var}(X)\) = EX^2 = 1.

Normal distribution#

Take \(\sigma >0, \mu\in\mathbb{R}\) and \(Y=\sigma X + \mu\) where \(X\) has a standard normal distribution, it can be computed that \(EY=\mu\) and \(\text{var}(Y) = \sigma^2\). Furthermore \(Y\) has density

\[(2\pi\sigma^2)^{-1/2}\exp(-(y-\mu)^2/2\sigma^2).\]

Bernoulli distribution#

We say that \(X\) has a Bernoulli distribution with parameter \(p\) if \(P(X=1)=p\) and \(P(X=0)=1-p\) and either \(X=1\) or \(X=0\) (actually \(X(\omega)=1\) or \(X(\omega)=0\)).

We have \(EX = p, EX^2=p\) and \(\text{var}(X) = p(1-p)\).

Poisson distribution#

We say that \(X\) has a Poisson distribution with parameter \(\lambda\) if

\[P(X=k) = e^{-\lambda}\lambda^k/k!\]

for \(k=0,1,2,\ldots\). \(EX=\lambda\), and \(\text{var}(X) = \lambda\). In general,

\[E(X(X-1\ldots(X-k+1))) = \lambda^k.\]

Geometric distribution#

\(N\) is said to have a geometric distribution with success probability \(p\in(0,1)\) if

\[P(N=k) = p(1-p)^{k-1}\]

for \(k=1,2,…\). \(N\) is the number of independent traials needed to observe an event with probability \(p\). We have

• \(EN = 1/p\),
• \(EN(N-1) = \frac{2(1-p)}{p^2}\),
• \(\text{var}(N) = \frac{1-p}{p^2}\).