Probability Distributions for Continuous Variables (PDF)

Recall from Two Types of Random Variables - Discretes vs. Continuous that an rv if continuous if:

A discrete random variable is a random variable (rv) whose possible values constitute either a finite or a countably infinite set (ex: the set of all integers, or the set of all positive integers).
A rv is continuous if both of the following apply:

Its set of possible values consists either of all $R$ or all numbers in a disjoint union of such intervals (ex: $[0, 10] \cup [20, 30]$ ).
No possible value of the variable has positive probability, namely $P (X = c) = 0$ $\forall c \in R$ .

Then we get the following:

Probability Distribution, Probability Density Function (pdf)

A probability distribution or probability density function (pdf) of a continuous random variable $X$ is a function $f (x)$ such that for any two numbers $a, b$ with $a \leq b$ :

P (a \leq X \leq b) = \int_{a}^{b} f (x) d x

Namely the probability between $(a, b)$ is the area under $f$ on that interval.

The following must be satisfied for a legitimate pdf:

$f (x) \geq 0$ for all $x$
$\int_{- \infty}^{\infty} f (x) d x = 1$

uniform distribution

A continuous rv $X$ has a uniform distribution on the interval $[A, B]$ if the pdf of $X$ is:

f (x; A, B) = \frac{1}{B - A} : A \leq x \leq B

We say then $X \sim Unif [A, B]$ .

Because $P (X = c) = 0$ via property 2 from the definition of continuity, then that means:

P (a \leq X \leq b) = P (a < X < b) = P (a \leq X < b) = P (a < X \leq b)

Namely the $<, \leq$ are equivalent here.

This only works for continuous variables! This isn't always true for the discrete case!