Random Variables

A random variable X is a function that maps the sample space $S$ to the real line;

for each element $w \in S, X (w) \in R$
(workable values are numbers)

Capital letters will usually be the random variable, and lowercase their realized value

Discrete Random Variables

$w \in S$	HHH	HHT	HTH	...
$P (w)$	$\frac{1}{8}$	$\frac{1}{8}$	$\frac{1}{8}$	...
$X (w)$	$3$	$2$	$2$	...
Each value of $X$ has a certain chance of happening

$X$	0	1	2	3
$P (X = x)$	$\frac{1}{8}$	$\frac{3}{8}$	$\frac{3}{8}$	$\frac{1}{8}$
For every $x$ outside of the domain $X$ , $P (X = x) = 0$

Continuous Random Variables

stats1

a random variables is a quantitative variable whose observed value is partly due to chance
- random variable x realized value
- random variable = X, Y, Z etc (theoretical quantity)
- realized value/realization of the random variable(x)
- P(X = x)
- P(X = 5%) for example
- tossing coins, sampling randomly
- sometimes we don't know the value of the variable until the experiment, but we may know characteristics of its distribution

random variables can be either discrete or continuous
- discrete = have a countable set of possible units
  - no in-between the values
    - yes or no, how many eggs broken
  - (n+1) makes sense
- continuous = have an infinite number of possible values
  - Continuous Random Variables
  - all values in an interval
    - weight, temperature, time
  - needs calculus