Statistical Distributions

This lesson introduces statistical distributions at A-Level, focusing on the concept of a random variable and its probability distribution. Understanding distributions is fundamental to modelling real-world phenomena and underpins all of inferential statistics.

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Random Variables

A random variable is a variable whose value depends on the outcome of a random event. Random variables are denoted by capital letters (e.g., $X$X), and their specific values by lowercase letters (e.g., $x$x).

Type	Description	Example
Discrete	Takes specific, countable values	Number of heads in 10 coin tosses
Continuous	Takes any value in a range	Height of a randomly chosen student

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Discrete Probability Distributions

A probability distribution for a discrete random variable $X$X lists all possible values and their probabilities.

Properties:

1. $0 \leq P(X = x) \leq 1$0≤P(X=x)≤1 for all values $x$x.
2. $\sum P(X = x) = 1$∑P(X=x)=1 (the probabilities of all possible values sum to 1).

Example:

$x$x	1	2	3	4
$P(X = x)$P(X=x)	0.1	0.3	0.4	0.2