Bernoulli Processes

Definition A random process is a collection of random variables $X_t$, one for each value of an index $t$ in an index set $\mathcal{T}$, all defined on the same sample space $\Omega$ with the same probability $P$. The index $t$ is called the time index. In discrete time the index set is a set of integers such as $\{1, 2, 3, \dots\}$, and in continuous time it is an interval such as $[0, \infty)$. For each fixed $t$ the value $X_t$ is one random variable. For each fixed outcome $\omega$ in $\Omega$, the values $X_t(\omega)$, viewed as a function of $t$, form a sample path of the process.

Definition Let $0 < p < 1$. A Bernoulli process with success probability $p$ is a sequence of random variables $X_1, X_2, \dots$ that are mutually independent and each have the $\operatorname{Bernoulli}(p)$ distribution, so $P(X_i = 1) = p$ and $P(X_i = 0) = 1 - p$ for every $i$. Together these conditions say that for every $n \ge 1$ and every choice of $x_1, \dots, x_n$ in $\{0, 1\}$,

where $k = x_1 + \cdots + x_n$ is the number of ones. The outcome coded $1$ is called a success and the outcome coded $0$ a failure.

Definition In a Bernoulli process with success probability $p$, the cumulative count after $n$ trials is

the number of successes among trials $1$ to $n$, with $C_0 = 0$. For each fixed $n \ge 1$ the count has the binomial distribution $C_n \sim \operatorname{Bin}(n, p)$, so for $k = 0, 1, \dots, n$

with $E[C_n] = np$ and $\operatorname{Var}(C_n) = np(1 - p)$. The sequence $C_0, C_1, C_2, \dots$ is itself a random process. It never decreases, and from one trial to the next it rises by $0$ or by $1$.

Definition In a Bernoulli process with success probability $p$, the first success time is

the number of the first trial that succeeds. The event $\{T = k\}$ requires failures on trials $1$ to $k - 1$ and a success on trial $k$, so by independence

for $k = 1, 2, 3, \dots$. This is the geometric distribution on $\{1, 2, 3, \dots\}$, and $E[T] = 1/p$. The event that no trial ever succeeds is contained in $\{C_n = 0\}$ for every $n$, so its probability is at most $(1 - p)^n$ for every $n$. It is therefore $0$, and $T$ is finite with probability $1$. Some books record instead the number of failures before the first success. That variable is $T - 1$, it takes the values $0, 1, 2, \dots$, and its mean is $(1 - p)/p$.

Definition For every $n \ge 1$, the first success time and the cumulative count satisfy the event identities

so $P(T > n) = (1 - p)^n$ and $P(T \le n) = 1 - (1 - p)^n$. Also $\{T = n\} = \{C_{n-1} = 0\} \cap \{X_n = 1\}$. For integers $m, n \ge 0$ the conditioning event $\{T > m\}$ has probability $(1 - p)^m > 0$, and since $\{T > m + n\}$ is contained in $\{T > m\}$,

This equals $P(T > n)$, which is the memoryless property of the geometric waiting time.

Definition A block of trials is a set of trial numbers, and blocks are disjoint when no trial belongs to two of them. In a Bernoulli process, the counts of successes in disjoint blocks are mutually independent, because each count is a function of the trials in its own block and the trials are mutually independent. A block of $m$ trials has a $\operatorname{Bin}(m, p)$ count wherever it lies. Cumulative counts overlap instead. For $1 \le m < n$,

where $C_m$ and the count $C_n - C_m$ of trials $m + 1$ to $n$ are independent. The cumulative counts $C_m$ and $C_n$ themselves are dependent, since