Geometric distribution (a discrete distribution)

Geometric distribution

Geometric distribution is a generalization of Bernoulli distribution , Repeated Bernoulli test , Until the first success , A random variable $X$ Indicates the number of tests completed at the time of first success , We call $X$ Is a random variable that obeys the geometric distribution

Screenshot of the source ：Geometric distribution

Here we introduce the first geometric distribution

Mean and variance
We use it ${\mu }_X$ Mean value
The first test was successful , The probability of success is $p$, Number of tests completed $X=1$, Expectations for this round of tests $1\cdot p$
The first test failed , Number of tests completed $X=1$, The expectation is $1\cdot (1-p)$, The experiment was restarted , front ${\mu }_X$ Test failed , Failure probability $1-p$, Number of tests completed $X={\mu }_X$, The expectation is ${\mu }_X(1-p)$, Expectations for this round of tests $1\cdot (1-p)+{\mu }_X(1-p)=({\mu }_X+1)(1-p)$

$$\begin{gathered} {\mu }_X=1\cdot p+({\mu }_X+1)(1-p) \\ p{\mu }_X=1 \\ {\mu }_X=\frac{1}{p} \end{gathered}$$

Screenshot of the source ：Geometric distribution

Example ：

Screenshot of the source ：Geometric distribution