7.3. Univariate Distributions

7.3.1. Gaussian Distribution

7.3.1.1. Standard Normal Distribution

This distribution has a mean of 0 and a variance of 1. It is denoted by

$$X\sim \mathcal{N}(0,1).$$

The PDF is given by

$$f_X(x)=\frac{1}{\sqrt{2\pi }}\exp (-\frac{x^2}{2}).$$

The CDF is given by

$$F_X(x)={\int }_{-\mathrm{\infty }}^x{f}_X(t)dt=\frac{1}{\sqrt{2\pi }}{\int }_{-\mathrm{\infty }}^x\exp (-\frac{t^2}{2})dt.$$

Symmetry

$$f(-x)=f(x).F(-x)+F(x)=1.$$

Some specific values

$$F_X(-\mathrm{\infty })=0,F_X(0)=\frac{1}{2},F_X(\mathrm{\infty })=1.$$

The Q-function is given as

$$Q(x)={\int }_x^{\mathrm{\infty }}{f}_X(t)dt=\frac{1}{\sqrt{2\pi }}{\int }_x^{\mathrm{\infty }}\exp (-\frac{t^2}{2})dt.$$

We have

$$F_X(x)+Q(x)=1.$$

Alternatively

$$F_X(x)=1-Q(x).$$

Further

$$Q(x)+Q(-x)=1.$$

This is due to the symmetry of normal distribution. Alternatively

$$Q(x)=1-Q(-x).$$

Probability of $X$ falling in a range $[a,b]$

$$\mathbb{P}(a\le X\le b)=Q(a)-Q(b)=F(b)-F(a).$$

The characteristic function is

$${\mathrm{\Psi }}_X(j\omega )=\exp (-\frac{{\omega }^2}{2}).$$

Mean:

$$\mu =\mathbb{E}(X)=0.$$

Mean square value

$$\mathbb{E}(X^2)=1.$$

Variance:

$${\sigma }^2=\mathbb{E}(X^2)-\mathbb{E}(X{)}^2=1.$$

Standard deviation

$$\sigma =1.$$

An upper bound on Q-function

$$Q(x)\le \frac{1}{2}\exp (-\frac{x^2}{2}).$$

The moment generating function is

$$M_X(t)=\exp \left(\frac{t^2}{2}\right).$$

7.3.1.2. Error Function

The error function is defined as

$$\mathrm{erf}(x)\triangleq \frac{2}{\sqrt{\pi }}{\int }_0^x\exp (-t^2)dt.$$

The complementary error function is defined as

$$\mathrm{erfc}(x)=1-\mathrm{erf}(x)=\frac{2}{\sqrt{\pi }}{\int }_x^{\mathrm{\infty }}\exp (-t^2)dt.$$

Error function is an odd function.

$$\mathrm{erf}(-x)=-\mathrm{erf}(x).$$

Some specific values of error function.

$$\mathrm{erf}(0)=0,\mathrm{erf}(-\mathrm{\infty })=-1,\mathrm{erf}(\mathrm{\infty })=1.$$

The relationship with normal CDF.

$$F_X(x)=\frac{1}{2}+\frac{1}{2}\mathrm{erf}\left(\frac{x}{\sqrt{2}}\right)=\frac{1}{2}\mathrm{erfc}(-\frac{x}{\sqrt{2}}).$$

Relationship with Q function.

$$Q(x)=\frac{1}{2}\mathrm{erfc}\left(\frac{x}{\sqrt{2}}\right)=\frac{1}{2}-\frac{1}{2}\mathrm{erf}\left(\frac{x}{\sqrt{2}}\right).$$

$$\mathrm{erfc}(x)=2Q(\sqrt{2}x).$$

We also have some useful results:

$${\int }_0^{\mathrm{\infty }}\exp (-\frac{t^2}{2})dt=\sqrt{\frac{\pi }{2}}.$$

7.3.1.3. General Normal Distribution

The general Gaussian (or normal) random variable is denoted as

$$X\sim \mathcal{N}(\mu ,{\sigma }^2).$$

Its PDF is

$$f_X(x)=\frac{1}{\sqrt{2\pi }\sigma }\exp (\frac{1}{2}\frac{(x-\mu {)}^2}{{\sigma }^2}.)$$

A simple transformation

$$Y=\frac{X-\mu }{\sigma }$$

converts it into standard normal random variable.

The mean:

$$\mathbb{E}(X)=\mu .$$

The mean square value:

$$\mathbb{E}(X^2)={\sigma }^2+{\mu }^2.$$

The variance:

$$\mathbb{E}(X^2)-\mathbb{E}(X{)}^2={\sigma }^2.$$

The CDF:

$$F_X(x)=\frac{1}{2}+\frac{1}{2}\mathrm{erf}\left(\frac{x-\mu }{\sigma \sqrt{2}}\right).$$

Notice the transformation from $x$ to $(x-\mu )/\sigma$.

The characteristic function:

$${\mathrm{\Psi }}_X(j\omega )=\exp (j\omega \mu -\frac{{\omega }^2{\sigma }^2}{2}).$$

Naturally putting $\mu =0$ and $\sigma =1$, it reduces to the CF of the standard normal r.v.

Th MGF:

$$M_X(t)=\exp (\mu t+\frac{{\sigma }^2{t}^2}{2}).$$

Skewness is zero and Kurtosis is zero.