Bernoulli distribution (a discrete distribution)

Bernoulli distribution

Note source ：Introduction to the Bernoulli Distribution

Bernoulli distribution binomial distribution （n Experiments ） The special case of , One experiment was carried out （$n=1$） The result of each experiment is only successful / Failure , It is also a special case of two-point distribution , The possible result need not be 0 and 1

Screenshot of the source ：Bernoulli distribution

The following figure shows three examples of Bernoulli distribution

$x=0,1$
$$\begin{gathered} P(X=0)=p^0(1-p{)}^{1-0}=1-p \\ P(X=1)=p^1(1-p{)}^{1-1}=p \end{gathered}$$
mean value

variance

Example ：

$$\begin{gathered} \text{The probability of randomly selecting a lawyer}p=\frac{1}{200} \\ \text{The probability of randomly selecting a person who is not a lawyer}p=1-\frac{1}{200}=\frac{199}{200} \\ P(X=x)=(\frac{1}{200}{)}^x(1-\frac{1}{200}{)}^{1-x} \\ \text{for x=0,1} \\ P(X=0)=(\frac{1}{200}{)}^0(1-\frac{1}{200}{)}^{1-0}=\frac{199}{200} \\ P(X=1)=(\frac{1}{200}{)}^1(1-\frac{1}{200}{)}^{1-1}=\frac{1}{200} \end{gathered}$$