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Bernoulli Distribution

In Probability,

11 trial​

  • 2 possible outcomes,
  • 1 trial
P(X=x)=Px(1−P)1−xP(X=x) = P^x (1-P)^{1-x}

where XX is the number of heads and x∈{0, 1}x \in \{0, ~1\}. PP is the probability of success.

Bernoulli Trials​

  • Independent. P(A1∩A2∩A3∩⋯∩An)=∏k=1nP(Ak)P(A_1 \cap A_2 \cap A_3 \cap \cdots \cap A_n) = \prod\limits_{k=1}^n P(A_k)
  • Stationary. Same PP.

nn trials​

  • 2 possible outcomes,
  • nn trial
P(X=x)=(nx)Px(1−P)1−xP(X=x) = {n \choose x} P^x (1-P)^{1-x}

Things to consider​

  • Number of Outcomes. Two or More?
  • With or Without Replacement?
StrategiesWith ReplacementWithout Replacement
22 outcomesBinomialHypergeometric
≥3\geq 3 outcomesMultinomialMultivariate hypergeometric

Multinomial​

kk- outcomes

N1N_1 = # of item 1 N2N_2 = # of item 2 N3N_3 = # of item 3 N4N_4 = # of item 4

...

NkN_k = # of item kk

N=N1+N2+N3+⋯+NkN = N_1 + N_2 + N_3 + \cdots + N_k
n=x1+x2+x3+⋯+xkn = x_1 + x_2 + x_3 + \cdots + x_k
P1=Probability(Item 1)P_1 = \text{Probability}(\text{Item 1})
Pk=Probability(Item k)P_k = \text{Probability}(\text{Item k})
P1+P2+P3+⋯+Pk=1P_1 + P_2 + P_3 + \cdots + P_k = 1
P(X1=x1, X2=x2,⋯ , Xk=xk)=n!x1! x2! x3! ⋯ xk!P1x1P2x2⋯PkxkP(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(X1=x1, X2=x2,⋯ , Xk=xk)=(N1x1)(N2x2)⋯(Nkxk)(Nn)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}}