Question

(5) There are $n$ eggs, each of which hatches a chick with probability $p$ (independently). Each of these chicks survives with probability $r$, independently. What is the distribution of the number of chicks that hatch? What is the distribution of the number of chicks that survive? (Give the PMFs; also give the names of the distributions and their parameters, if applicable.)

Name: Problem 28, Chapter 3: Introduction to Probability
Uploaded: 2022-11-08T00:53:59-08:00
Channel: Victor Salazar
Description: (5) There are n eggs, each of which hatches a chick with probability p (independently). Each of these chicks survives with probability r, independently. What is the distribution of the number of chicks that hatch? What is the distribution of the number of chicks that survive? (Give the PMFs; also give the names of the distributions and their parameters, if applicable.)

   (5) There are $n$ eggs, each of which hatches a chick with probability $p$ (independently). Each of these chicks survives with probability $r$, independently. What is the distribution of the number of chicks that hatch? What is the distribution of the number of chicks that survive? (Give the PMFs; also give the names of the distributions and their parameters, if applicable.)

Introduction to Probability

Joseph K.… 2nd Edition

Chapter 3, Problem 28

Instant Answer

Solved by Expert Victor Salazar

11/08/2022

Step 1

Each egg hatches with probability $p$, independently of the others. This is a classic example of a binomial distribution, with $n$ trials (eggs) and success probability $p$. So, the number of chicks that hatch follows a Binomial distribution with parameters $n$ Show more…

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(5) There are $n$ eggs, each of which hatches a chick with probability $p$ (independently). Each of these chicks survives with probability $r$, independently. What is the distribution of the number of chicks that hatch? What is the distribution of the number of chicks that survive? (Give the PMFs; also give the names of the distributions and their parameters, if applicable.)

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Key Concepts

Binomial Distribution

The binomial distribution describes the probability of having a fixed number of successes in a fixed number of independent trials, where each trial has two outcomes (success or failure) and the probability of success remains constant. The probability mass function (PMF) is given by P(X = k) = C(n, k) p^k (1 - p)^(n - k), where n is the number of trials and p is the probability of success in each trial.

Compound Binomial Process

A compound binomial process occurs when a two-stage experiment is performed in which the outcome of one binomial process influences a second independent binomial process. In such a scenario, each trial undergoes an initial test with probability p of success and, if successful, a further test with an independent probability r of success. The overall effect is that the final successful outcome (a success in both stages) has probability p multiplied by r, leading to a binomial distribution with parameters n and pr.

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