An event independently occurs on each day with probability...

An event independently occurs on each day with probability $p$. Let $N(n)$ denote the total number of events that occur on the first $n$ days, and let $T_{r}$ denote the day on which the $r$ th event occurs. (a) What is the distribution of $N(n) ?$ (b) What is the distribution of $T_{1} ?$ (c) What is the distribution of $T_{r} ?$ (d) Given that $N(n)=r$, show that the unordered set of $r$ days on which events occurred has the same distribution as a random selection (without replacement) of $r$ of the values $1,2, \ldots, n$.

An event independently occurs on each day with probability $p$. Let $N(n)$ denote the total number of events that occur on the first $n$ days, and let $T_{r}$ denote the day on which the $r$ th event occurs.
(a) What is the distribution of $N(n) ?$
(b) What is the distribution of $T_{1} ?$
(c) What is the distribution of $T_{r} ?$
(d) Given that $N(n)=r$, show that the unordered set of $r$ days on which events occurred has the same distribution as a random selection (without replacement) of $r$ of the values $1,2, \ldots, n$.

Key Concepts

Random Sampling Without Replacement

When conditioning on the total number of events (e.g., N(n)=r), the specific days on which events occur are uniformly random among all possible combinations of r days out of n. This concept of random sampling without replacement shows that, given the fixed count of events, the unordered set of days is distributed just like a random selection of r days from 1 to n.

Geometric Distribution

The geometric distribution models the number of Bernoulli trials needed to achieve the first success. In the context of the problem, the day on which the first event occurs (denoted as T1) is distributed geometrically with parameter p, representing the waiting time until the first event.

Negative Binomial Distribution

The negative binomial distribution generalizes the geometric distribution to model the number of trials needed to achieve a given number of successes. Here, the day on which the rth event occurs (denoted as Tr) follows a negative binomial distribution with parameters corresponding to the target number of successes r and the success probability p.

Bernoulli Trials

In a Bernoulli trial framework, each day is treated as an independent trial with two outcomes – either an event occurs (success) with probability p or does not occur (failure) with probability 1-p. This is the foundation of the process described in the problem, where the same probability p applies to each day independently.

Binomial Distribution

The binomial distribution arises when counting the number of successes in a fixed number of independent Bernoulli trials. In the problem, the total number of events in the first n days (denoted as N(n)) follows a binomial distribution, with parameters n (number of trials) and p (probability of success on each trial).

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