Bernoulli Process

Topics: Stochastic Process- Bernoulli Trial

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(definition)

A stochastic process $\left\{X_n\mid n\in \mathbb{N}\right\}$ is called a Bernoulli process if it satisfies the following conditions:

1. $X_1,X_2,\dots$ are independent
2. $\mathbb{P}(X_n=1)=p$, $\mathbb{P}(X_n=0)=1-p=q$

We’ll refer to the event $X_n=1$ as success and to $X_n=0$ as failure.

Number of Successes §

(definition)

Let $\left\{X_n\mid n\in \mathbb{N}\right\}$ be a Bernoulli process with a success probability of $p$.

The number of successes we’ve had up to the $n$th trial is denoted by $N_n$ and defined as:

$$N_n=\{\begin{array}{ll}0& \text{if }n=0\\ X_1+\cdots +X_n& \text{if }n=1,2,3,\dots \end{array}$$

Note that $N_n$ defines another stochastic process, one with discrete state and parameter sets.

(observation)

From the previous definition, we can tell that $N_{n+m}-N_n$ is the number of successes we’ve had along the trials $n+1,n+2,\dots ,n+m$.

Distribution of $X_n$ and $N_n$ §

(theorem)

Each of these individual $X_n$ follow a Bernoulli distribution with a parameter $p$.

As such, $N_n$ follows a binomial distribution with parameters $n,p$, since the sum of independent Bernoulli-distributed random variables distributes binomially.

(theorem)

Now, we can tell that for $n\in \mathbb{N}$:

1. $\mathbb{P}(N_n=k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k{q}^{n-k}k=0,1,\dots ,n$
2. $\mathbb{E}[N_n]=np$
3. $\mathrm{Var}(N_n)=npq$

Basically what was already established for a Bernoulli distribution.