Probability distribution

The probability distribution of a random variable $X$ is the function that describes the probabilities for the set of possible values. The probability mass function of a discrete random variable is the function $p(x)=P(X=x)$ that describes the probabilities for the set of possible values.

The probability density function (PDF) describes a continuous random variable’s $X$ probability (since its probabilities are given by the area under the curve). For PDFs:

$${\int }_{-\infty }^{\infty }f(x)dx=1$$

As an aside: we can rigorously build up random variables and probability distributions with the idea of measure theory.

Fundamentals

The cumulative distribution function specifies the probability that a random variable is less than or equal to a given value, given by:

$$F(x)=P(X\le x)={\int }_{-\infty }^{x}f(t)dt$$

The mean (or expected value) of $X$ is:

$${\mu }_X=E(X)={\int }_{-\infty }^{\infty }xf(x)dx$$

The variance is:

$${\sigma }_X^2=\text{Var}(X)={\int }_{-\infty }^{\infty }(x-{\mu }_X{)}^{2}f(x)dx={\int }_{-\infty }^{\infty }{x}^{2}f(x)dx-{\mu }_X^2$$

The median is the point $x_m$ where:

$$F(x_m)={\int }_{-\infty }^{x_m}f(x)dx=0.5$$

The $p$-th percentile (where $p\in [0,100]$) is the point $x_p$ where:

$$F(x_p)={\int }_{-\infty }^{x_p}f(x)dx=\frac{p}{100}$$

The median is the 50th percentile.

Types

• Bernoulli distribution
• Binomial distribution
• Poisson distribution
• Normal distribution