Common Continuous Distributions

A random variable $X$ has the uniform distribution on $[a, b]$, written $X \sim \text{Uniform}[a, b]$, if its PDF is

$$f(x) = \frac{1}{b - a}\,\mathbf{1}_{\{x \in [a,b]\}}.$$

The CDF is $F(x) = \frac{x - a}{b - a}$ for $x \in [a, b]$. Mean: $(a+b)/2$. Variance: $(b-a)^2/12$.

A random variable $X$ has the exponential distribution with rate $\lambda > 0$, written $X \sim \text{Exp}(\lambda)$, if its PDF is

$$f(x) = \lambda e^{-\lambda x}\,\mathbf{1}_{\{x \geq 0\}}.$$

CDF: $F(x) = 1 - e^{-\lambda x}$ for $x \geq 0$. Mean: $1/\lambda$. Variance: $1/\lambda^2$.

A random variable $X$ has the Gaussian (or normal) distribution with mean $\mu$ and variance $\sigma^2 > 0$, written $X \sim \mathcal{N}(\mu, \sigma^2)$, if its PDF is

$$f(x) = \frac{1}{\sqrt{2\pi\sigma^2}}\exp\!\left(-\frac{(x - \mu)^2}{2\sigma^2}\right).$$

The standard normal distribution is $\mathcal{N}(0, 1)$.

A random variable $X$ has the Gamma distribution with shape $\alpha > 0$ and rate $\lambda > 0$, written $X \sim \text{Gamma}(\alpha, \lambda)$, if its PDF is

$$f(x) = \frac{\lambda^\alpha}{\Gamma(\alpha)}\,x^{\alpha - 1}e^{-\lambda x}\,\mathbf{1}_{\{x \geq 0\}},$$

where $\Gamma(\alpha) = \int_0^{\infty} t^{\alpha-1}e^{-t}\,dt$.

Mean: $\alpha/\lambda$. Variance: $\alpha/\lambda^2$.

A random variable $X$ has the Beta distribution with parameters $a, b > 0$, written $X \sim \text{Beta}(a, b)$, if its PDF is

$$f(x) = \frac{1}{B(a,b)}\,x^{a-1}(1-x)^{b-1}\,\mathbf{1}_{\{x \in [0,1]\}},$$

where $B(a,b) = \Gamma(a)\Gamma(b)/\Gamma(a+b)$.

Mean: $a/(a+b)$. Variance: $ab/[(a+b)^2(a+b+1)]$.

A random variable $T$ has the Student-$t$ distribution with $\nu$ degrees of freedom, written $T \sim t_\nu$, if its PDF is

$$f(t) = \frac{\Gamma\!\left(\frac{\nu+1}{2}\right)}{\sqrt{\nu\pi}\,\Gamma\!\left(\frac{\nu}{2}\right)}\left(1 + \frac{t^2}{\nu}\right)^{-(\nu+1)/2}.$$

Mean: $0$ (for $\nu > 1$). Variance: $\nu/(\nu - 2)$ (for $\nu > 2$).

If $X, Y \sim \mathcal{N}(0, \sigma^2)$ are independent, then $R = \sqrt{X^2 + Y^2}$ has the Rayleigh distribution with parameter $\sigma$:

$$f_R(r) = \frac{r}{\sigma^2}\exp\!\left(-\frac{r^2}{2\sigma^2}\right)\,\mathbf{1}_{\{r \geq 0\}}.$$

Mean: $\sigma\sqrt{\pi/2}$. Variance: $(2 - \pi/2)\sigma^2$.

If $X \sim \mathcal{N}(\mu_1, \sigma^2)$ and $Y \sim \mathcal{N}(\mu_2, \sigma^2)$ are independent with $s = \sqrt{\mu_1^2 + \mu_2^2}$ (the line-of-sight amplitude), then $R = \sqrt{X^2 + Y^2}$ has the Ricean distribution with parameters $s$ and $\sigma$:

$$f_R(r) = \frac{r}{\sigma^2}\exp\!\left(-\frac{r^2 + s^2}{2\sigma^2}\right)I_0\!\left(\frac{rs}{\sigma^2}\right)\,\mathbf{1}_{\{r \geq 0\}},$$

where $I_0$ is the zeroth-order modified Bessel function of the first kind. The K-factor $K = s^2/(2\sigma^2)$ quantifies the ratio of direct to scattered power. When $K = 0$, the Ricean reduces to the Rayleigh.

A random variable $R$ has the Nakagami-$m$ distribution with shape $m \geq 1/2$ and spread $\Omega > 0$ if

$$f_R(r) = \frac{2m^m}{\Gamma(m)\,\Omega^m}\,r^{2m-1}\exp\!\left(-\frac{m r^2}{\Omega}\right)\,\mathbf{1}_{\{r \geq 0\}}.$$

For $m = 1$, this is the Rayleigh distribution with $\Omega = 2\sigma^2$. The Nakagami-$m$ is more flexible than the Rayleigh or Ricean because it can model a wider range of fading severities through the single parameter $m$.