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Definition: A random variable (or a distribution) is a numerical value
associated with an [[Experiment]] whose value can change from one replicate of
the experiment to another.
A proper definition would need [[Probability space]] and [[Measurable function]]s.
For example, given a fair dice roll,
$$X = \{1,2,3,4,5,6\}$$
is a random variable.
$$Y = [30, 260]$$
is another.
Two types: Discrete random variables and continuous random variables.
# [[Discrete]] random variable
If the outcomes are either bounded in size or countably infinite.
## Probability mass function
Also known as the pmf. Every discrete random variable has a corresponding pmf.
Denoted
$$p(x) = P(X = x)$$
## Table
Every discrete random variable also has a corresponding table.
|$X$|$x_1$|$x_2$|$...$|$x_n$|
|--|--|--|--|--|
|$p(x)$|$p(x_1)$|$p(x_2)$|$...$|$p(x_n)$|
where
$$p(x_i) \ge 0 \quad \forall i \in \{1,2,..,n\}$$
$$\sum_{i=1}^n p(x_i) = 1$$
# [[Continuous]] random variable
The rest. E.g. some interval on the number line.
## Probability density function
Also knows as the pdf. Every continuous random variable has a corresponding pdf.
Denoted $f(x)$ where
$$\int_a^b f(x) dx = P(a \le X \le b)$$
and
$$f(x) \le 0 \quad \forall x$$
$$\int_{-\infty}^\infty f(x) dx = 1$$
# Cumulative distribution function
Also knows as the cdf.
$$F(x) = P(X \le x)$$
For discrete random variables:
$$F(y) = \sum_{i=1}^y p(x_i)$$
And for continuous random variables:
$$F(x) = \int_{-\infty}^x f(y) dy$$
Here we see that
$$F'(x) = f(x)$$
for continuous random variables. Compare with [[Algebrans fundamentalsats]]?
# Examples
## Waiting time (useful model)
Let $X$ be the waiting time between calls in a phone center. Assume $X$ is a
continuous random variable with pdf
$$f(x) = 2e^{-2x} \quad x \gt 0$$
What is $P(X \gt 3)$?
$$P(X \gt 3) = \int_3^\infty f(x) dx = \int_3^\infty 2e^{-2x}dx = e^{-6}$$
In actuality,
$$f(x) = 2e^{-2x} \quad x>0$$
$$0 \ \mathrm{otherwise}$$
but the 0-case is assumed.
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