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Problem 2: A stochastic process \{N(t), t\geq 0\} is called a counting process if $N(t)$ represents the total number of events that have occured up to time $t$ and must satisfy: (i) $N(t) \geq 0$. (ii) $N(t)$ is integer valued. (iii) If $s < t$, then $N(s) \leq N(t)$. (iv) For $s < t$, $N(t) - N(s)$ equals the number of events that have occurred in the interval $(s, t]$. Now the counting process \{N(t), t\geq 0\} is said to be a Poisson process with rate $\lambda$, $\lambda > 0$, if (i) $N(0) = 0$. (ii) The process has independent increments. (iii) The number of events in an interval $t$ is Poisson distributed with mean $\lambda t$ and $\forall s, t \geq 0$ $P(N(t+s) - N(s) = n) = e^{-\lambda t} \frac{(\lambda t)^n}{n!}$ Based on the above definitions, now consider a Poisson process and let $X_1$ denote the time of first event arrival. Let $X_n$, $n \geq 1$, denotes the time between $(n-1)^{st}$ and $n^{th}$ events. The sequence \{$X_n$, $n \geq 1$\} is called sequence of interarrival times. Find the probability distribution of $X_n$.

          Problem 2: A stochastic process \{N(t), t\geq 0\} is called a counting process if $N(t)$ represents the total number of events that have occured up to time $t$ and must satisfy:
(i) $N(t) \geq 0$.
(ii) $N(t)$ is integer valued.
(iii) If $s < t$, then $N(s) \leq N(t)$.
(iv) For $s < t$, $N(t) - N(s)$ equals the number of events that have occurred in the interval $(s, t]$.
Now the counting process \{N(t), t\geq 0\} is said to be a Poisson process with rate $\lambda$, $\lambda > 0$, if
(i) $N(0) = 0$.
(ii) The process has independent increments.
(iii) The number of events in an interval $t$ is Poisson distributed with mean $\lambda t$ and $\forall s, t \geq 0$
$P(N(t+s) - N(s) = n) = e^{-\lambda t} \frac{(\lambda t)^n}{n!}$
Based on the above definitions, now consider a Poisson process and let $X_1$ denote the time of first event arrival. Let $X_n$, $n \geq 1$, denotes the time between $(n-1)^{st}$ and $n^{th}$ events. The sequence \{$X_n$, $n \geq 1$\} is called sequence of interarrival times. Find the probability distribution of $X_n$.

$Problem 2: A stochastic process {N(t), t≥0} is called a counting process if N(t) represents the total number of events that have occured up to time t and must satisfy: (i) N(t) ≥ 0. (ii) N(t) is integer valued. (iii) If s < t, then N(s) ≤ N(t). (iv) For s < t, N(t) - N(s) equals the number of events that have occurred in the interval (s, t]. Now the counting process {N(t), t≥0} is said to be a Poisson process with rate λ, λ > 0, if (i) N(0) = 0. (ii) The process has independent increments. (iii) The number of events in an interval t is Poisson distributed with mean λ t and ∀ s, t ≥ 0 P(N(t+s) - N(s) = n) = e^-λ t((λ t)^n)/(n!) Based on the above definitions, now consider a Poisson process and let X1 denote the time of first event arrival. Let Xn, n ≥ 1, denotes the time between (n-1)^st and n^th events. The sequence {Xn, n ≥ 1} is called sequence of interarrival times. Find the probability distribution of Xn.$

Added by Bernard M.

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