A statistical experiment is any process where we obtain measurements, for example, measuring the rainfall (in inches). If the value being measured is the result of chance or random outcome, then it’s called a random variable.
Distribution: Possible values a variable can take, and how frequently they occur.
$x$ denotes all possible outcomes
$X$ denotes the “actual” outcome of the event
$P(X=x)$ denotes the likelihood of reaching a particular outcome
Example: X = number of red marbles we draw out of a bag, then P(X=5) denotes probability of getting 5 red marbles.
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Since $P(X=x)$ denotes the probability of each distinct outcome, it’s called a Probability Function.
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Distributions are defined using two characteristics: mean and variance.
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A probability distribution lists the probability of each outcome.
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For example, below table is the probability distribution of rolling a fair dice:
| Outcome | Probability |
|---|---|
| $P(X=1)$ | 1/6 |
| $P(X=2)$ | 1/6 |
| $P(X=3)$ | 1/6 |
| $P(X=4)$ | 1/6 |
| $P(X=5)$ | 1/6 |
| $P(X=6)$ | 1/6 |