1.1 Probability distributions

The distribution $P_X$ of a random variable $X$ is simply a probability measure which assigns probabilities to events on thereal line. The distribution $P_X$ answers questions of the form:

What is the probability that $X$ lies in some subset $F$ of the real line?

In practice we summarize $P_X$ by its Probability Mass Function - pmf (for discrete variables only), Probability Density Function - pdf (mainly for continuous variables), or Cumulative Distribution Function - cdf (for either discrete or continuous variables).

Probability mass function (pmf)

Suppose the discrete random variable $X$ can take a set of $M$ real values $\{x_1, \dots , x_M\}$ , then the pmf is defined as:

Cumulative distribution function (cdf)

The cdf can describe discrete, continuous or mixed distributions of $X$ and is defined as:

Properties follow directly from the Axioms of Probability:

• $0\le F_X(x)\le 1$
• $F_X(())$∞ 0 , $F_X()$∞ 1
• $F_X(x)$ is non-decreasing as $x$ increases
• $(x_1< X\le x_2)=F_X(x_2)-F_X(x_1)$
• $(X> x)=1-F_X(x)$

Probability density function (pdf)

The pdf of $X$ is defined as the derivative of the cdf:

The pdf and cdf of a continuous distribution (in this case the normal or Gaussian distribution) are shown in (d) and (e) of .

Properties of pdfs:

• $f_X(x)\ge 0$
• $\int_{()} \,d x$∞ ∞ f X x 1
• $F_X(x)=\int_{()} \,d \alpha$∞ x f X α
• $(x_1< X\le x_2)=\int_{x_1}^{x_2} f_X(\alpha )\,d \alpha$