Bernoulli Distribution

In Probability,

$1$ trial

• 2 possible outcomes,
• 1 trial

$$P(X=x) = P^x (1-P)^{1-x}$$

where $X$ is the number of heads and $x \in {0, ~1}$. $P$ is the probability of success.

Bernoulli Trials

• Independent. $P(A_1 \cap A_2 \cap A_3 \cap \cdots \cap A_n) = \prod\limits_{k=1}^n P(A_k)$
• Stationary. Same $P$.

$n$ trials

• 2 possible outcomes,
• $n$ trial

$$P(X=x) = {n \choose x} P^x (1-P)^{1-x}$$

Things to consider

• Number of Outcomes. Two or More?
• With or Without Replacement?

| Strategies | With Replacement | Without Replacement | | ----------------- | ------------ | --------------------------- | ------------------- | | $2$ outcomes | Binomial | Hypergeometric | | $\geq 3$ outcomes | Multinomial | Multivariate hypergeometric |

Multinomial

$k$- outcomes

$N_1$ = # of item 1 $N_2$ = # of item 2 $N_3$ = # of item 3 $N_4$ = # of item 4

...

$N_k$ = # of item $k$

$$N = N_1 + N_2 + N_3 + \cdots + N_k$$ $$n = x_1 + x_2 + x_3 + \cdots + x_k$$ $$P_1 = \text{Probability}(\text{Item 1})$$ $$P_k = \text{Probability}(\text{Item k})$$ $$P_1 + P_2 + P_3 + \cdots + P_k = 1$$ $$P(X_1 = x_1, ~ X_2 = x_2, \cdots , ~X_k=x_k) = {n! \over {x_1!~x_2!~x_3!~\cdots~x_k!}} P_1^{x_1} P_2^{x_2} \cdots P_k ^{x_k}$$ $$P(X_1 = x_1, ~ X_2 = x_2, \cdots , ~X_k=x_k) = {{{N_1 \choose x_1} {N_2 \choose x_2} \cdots {N_k \choose x_k}} \over {N \choose n}}$$