2. Probability distributions

Probability distributions

Experiment

An experiment is a procedure that yields one of a given set of possible outcomes.

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Sample space

The sample space, $S/\Omega$, of the experiment is the set of all possible outcomes $$ S/\Omega = \{ s_1, s_2, ..., s_n \} $$

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Probability distribution

A probability distribution in the sample space $S$ is a function $p$ assigning a number $p(s_i)$ to each possible outcome $s_i$ in $S$ (the probability of $s_i$), such that:

• values are exhaustive (cover everything)
• values are mutually exclusive (no overlap)
• $0\le p(s_i)\le 1$, for all $i=1\text{ to }n$
• $p(s_1)+p(s_2)+...+p(s_n)=1$ All probability values add up to $1$.

$$P(\text{Weather})=\left(\begin{array}{c}0.72\\ 0.1\\ 0.08\\ 0.1\end{array}\right)=⟨0.72,0.1,0.08,0.1⟩$$

This means, for example: $P(\text{Weather}=sunny)=0.72$.

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Uniform distribution

A uniform distribution is where the probabilities are evenly distributed across the sample space. $$ p(s_i) = \frac{1}{n} \text{ for all } i = 1 \text{ to } n $$

title: Example
Given the uniform distribution on $S$.
If a fair dice is rolled, then there are six possible equally likely outcomes: $1,2,3,4,5,6$.
So $p(1) = p(2) = ... = p(6) = \frac{1}{6}$.

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