<aside> <img src="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/22ebcca6-13c4-48f7-97e0-4da7593ce456/Probability_Distribution.png" alt="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/22ebcca6-13c4-48f7-97e0-4da7593ce456/Probability_Distribution.png" width="40px" /> Probability distribution: encompasses all the possible values of a random variable together with an indication of the probability that each of those values occurs.
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<aside> <img src="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/cf42273f-cff4-479b-9904-9611112942f2/Discrete_random_variable.png" alt="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/cf42273f-cff4-479b-9904-9611112942f2/Discrete_random_variable.png" width="40px" /> Discrete random variables: $X$ is a discrete random variable which can have a range of values given as $\{x_1,x_2,\cdots,x_i\}$ and $X$ maps the sample space, $S$ to real numbers $\R$:
$$ X:S\rightarrow \R $$
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$$ 0\le p_i\le 1 \,\forall i\quad\sum_i p_i=1 \quad P(X\in A)=\sum_{i\in A}p_i \,\text{where $A$ is a subset} $$
$$ \bar X\equiv E(X) \equiv \left< X\right >=\sum_ix_ip_i $$
$$ \left< g(X)\right>=\sum_ig(x_i)p_i $$
$$ \sigma^2=\left<X^2 \right>-\left< X\right>^2 $$
$$ \sigma=\sqrt{\sigma^2} $$
<aside> <img src="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/8425e046-953e-4057-9e7d-5779b96e999b/geometric_distribution.png" alt="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/8425e046-953e-4057-9e7d-5779b96e999b/geometric_distribution.png" width="40px" /> Geometric distribution:
$$ p_n=pq^n=p(1-p)^n\text{ for }n=0,1,2,3,... $$
$$ p\sum^\infin_{n=0}q^n=p+pq+pq^2+\cdots=\frac{p}{1-q}=1 $$
