Bernoulli Distribution

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cho.sh

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Bernoulli Distribution · cho.sh

Bernoulli Distribution

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In Probability,

$1$ trial

2 possible outcomes,
1 trial

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

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Comments

−

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?

[[Strategy	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}}

P1x1

P2x2

⋯

Pkxk

)

(x2N2)

⋯

(xkNk)

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Bernoulli Distribution

111 trial

Bernoulli Trials

nnn trials

Things to consider

Multinomial

$1$ trial

$n$ trials