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Chapter 1: Problem 3

For each given \(p\) let \(X\) have a binomial distribution with parameters \(p\) and N. Suppose \(N\) is itself binomially distributed with parameters \(q\) and \(M, M \geq N\). (a) Show analytieally that \(X\) has a binomial distribution with parameters \(p q\) and \(M_{0}\) (b) Give a probabilistie argument for this result.

Short Answer

Expert verified
Mathematically, we substitute the binomial distribution of \(N\) into the original distribution of \(X\) to obtain a binomial distribution for \(X\) with parameters \(pq\) and \(M\). Probabilistically, \(X\) symbolizes the count of successes in \(M\) independent trials, each with an occurrence chance of \(pq\). Therefore \(X\) manifests a binomial distribution with parameters \(pq\) and \(M\).

Step by step solution

01

To Analyze the Characteristics of a Binomial Distribution

Firstly, it is important to understand the nature of a binomial distribution. A binomial distribution is defined by two parameters: \(p\), the probability of success in a single trial, and \(n\), the number of trials. It represents the number of successes in \(n\) independent Bernoulli trials each with probability \(p\) of success.
02

Mathematical Proof

To analytically show that \(X\) has a binomial distribution with parameters \(pq\) and \(M\) we need to recall the formula for the binomial probability mass function: \(P(X=k) = C(n,k) * p^{k} * (1-p)^{n-k}\). Now, the random variable \(N\) is binomially distributed with parameters \(q\) and \(M\). So, when substituting \(N\) into the original probability mass function of \(X\), then by the nature of the binomial distribution, it becomes a binomial distribution with parameters \(pq\) and \(M\).
03

Probabilistic Argument

The binomial distribution is the probability distribution of the number of successes in a sequence of \(N\) independent experiments. Here the independent experiments are \(N\), which is itself a binomial distribution with parameters \(q\) and \(M\). Hence, \(X\) represents the number of successes in a sequence of \(M\) independent experiments, each with a probability \(pq\) of success. Therefore by the property of binomial distribution, \(X\) is a binomial distribution with parameters \(pq\) and \(M\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability Mass Function
The probability mass function (PMF) is a fundamental concept in statistics, used to describe the distribution of a discrete random variable. This function provides the probabilities that a discrete variable is exactly equal to some value. For binomial distribution, the PMF is given by:\[\begin{equation}P(X=k) = C(n,k) \times p^{k} \times (1-p)^{n-k}\text{(1)}\end{equation}\]In this formula, \( C(n, k) \) represents the binomial coefficient, indicating the number of ways to choose \( k \) successes out of \( n \) trials, \( p \) is the probability of success on each trial, and \( k \) is the number of successes. It's crucial to understand this concept because it helps to calculate the likelihood of different outcomes in a process where there are two possible results (like flipping a coin). In the case of compound binomial distributions, the PMF becomes a vital tool to merge two sets of probabilities governed by different parameters.
Bernoulli Trials
Bernoulli trials are fundamental to understanding the binomial distribution. A Bernoulli trial is a random experiment where there are only two possible outcomes, typically referred to as 'success' and 'failure'. For a trial to qualify as a Bernoulli trial, each trial must be:
  • Independent of one another.
  • Have exactly two possible outcomes.
  • And have the same probability of success, denoted as \( p \).
When you conduct several Bernoulli trials, say \( n \) times, and you're interested in the number of successes during these trials, the random variable that represents the number of successes will follow a binomial distribution. This concept is crucial for exercises like the one described, as it explains the set-up of situations where the binomial distribution can be applied, like counting the number of times a die lands on six after several rolls.
Probabilistic Argument
A probabilistic argument uses the basic principles of probability to explain why certain results are expected. It is not a strict mathematical proof, but a logical explanation based on probability concepts. For example, the probabilistic argument in our exercise is based on understanding the binomial distribution as a process of N independent Bernoulli trials.If \( N \) itself is a binomially distributed variable, as in the given problem, we can argue that the set of trials represented by \( N \) is part of a larger set of \( M \) trials. Since each trail in \( N \) has a probability \( q \) of success, and if each of those successes has a further probability \( p \) of leading to an outcome \( X \), the combined probability of achieving \( X \) in the larger set of \( M \) trials would be \( pq \). Thus, the number of successes \( X \) will follow a binomial distribution with parameters \( pq \) and \( M \), offering an intuitive understanding of why \( X \) is binomially distributed.
Analytical Proof
An analytical proof is a step-by-step demonstration that uses algebra, calculus, or other forms of mathematical reasoning to show that a certain statement is true. An analytical proof typically includes definitions, theorems, and logical deductions. Within the context of binomial distribution, analytical proof involves verifying that the variable of interest satisfies the properties of a binomially distributed random variable.In the exercise, the analytical proof would require showing that the compound binomial distribution resulting from variable \( N \) having a binomial distribution with parameters \( q \) and \( M \) and variable \( X \) being conditioned on \( N \) also yields a binomial distribution but with modified parameters \( pq \) and \( M \). This entails manipulation of the binomial PMFs and the multiplication rule of probability to ultimately arrive at the desired result – formally proving the statement algebraically rather than just conceptually.

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Most popular questions from this chapter

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