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Chapter 3: Problem 112

The negative binomial rv \(X\) was defined as the number of \(F^{\prime}\) s preceding the \(r\) th \(S\). Let \(Y=\) the number of trials necessary to obtain the rth \(S\). In the same manner in which the pmf of \(X\) was derived, derive the pmf of \(Y\).

Short Answer

Expert verified
The pmf of \( Y \) is \( P(Y = k) = \binom{k-1}{r-1} p^r (1-p)^{k-r} \).

Step by step solution

01

Understand the Problem

The problem asks us to derive the probability mass function (pmf) of a random variable \( Y \), which represents the total number of trials needed to get the \( r \)th desired outcome (success \( S \)) in a sequence of independent Bernoulli trials with success probability \( p \).
02

Define the Random Variable

The random variable \( Y \) represents the number of trials needed to achieve the \( r \)th success. It implies that in these \( Y \) trials, exactly \( r-1 \) successes have occurred in any of the first \( Y - 1 \) trials, and the \( Y \)th trial is a success.
03

Use Binomial Coefficients and Probabilities

To derive the pmf, we consider \( Y = k \) for \( k \geq r \). In the first \( k-1 \) trials, there need to be exactly \( r-1 \) successes. The probability of this happening is given by the binomial coefficient: \( \binom{k-1}{r-1}p^{r-1}(1-p)^{k-r} \).
04

Ensure the Last Trial is a Success

The \( k \)th trial must be a success for \( Y \) to equal \( k \). Thus, we multiply the probability from Step 3 by \( p \), the probability of having a success in trial \( k \).
05

Combine to Find the PMF

Combining what we derived in Steps 3 and 4, the pmf of \( Y \) is: \[ P(Y = k) = \binom{k-1}{r-1} p^r (1-p)^{k-r} \] This is valid for \( k = r, r+1, r+2, \ldots \).
06

Finalize the Expression

After simplifications, the expression is the negative binomial distribution. It describes the probability of needing \( k \) trials to achieve \( r \) successes in a series of Bernoulli trials.

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Key Concepts

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

Probability Mass Function (PMF)
The Probability Mass Function (PMF) is a vital concept in understanding discrete random variables. It is a function that gives the probability that a discrete random variable is exactly equal to some value.
When dealing with discrete distributions like the negative binomial, the PMF provides a complete description of the distribution鈥檚 probabilities.
  • For a random variable, say, \( Y \), the PMF is denoted by \( P(Y = k) \), which represents the probability that the random variable \( Y \) is equal to \( k \).
  • The sum of all probabilities in a PMF must equal 1, as they collectively represent all possible outcomes.
In our example, the PMF is crafted to represent the total number of trials needed (including the last successful one) to achieve a specified number of success, reflecting characteristics typical of the negative binomial distribution.
Bernoulli Trials
Bernoulli trials are a sequence of experiments or processes that have exactly two possible outcomes: success or failure. Named after the mathematician Jacob Bernoulli, these trials are fundamental building blocks in probability theory, especially in binomial and negative binomial distributions.
  • Every trial is independent, meaning the outcome of one trial does not affect the others.
  • Each trial has the same probability of success, denoted by \( p \).
Understanding Bernoulli trials is crucial in scenarios where repeated, independent experiments are conducted, similar to flipping a coin or rolling a die. In the negative binomial distribution, Bernoulli trials help determine the distribution of successes over multiple attempts.
Binomial Coefficient
The binomial coefficient is a mathematical concept used to determine the number of ways to choose a subset of elements from a larger set, without considering the order of selection. It's denoted as \( \binom{n}{k} \), which calculates the number of ways to choose \( k \) successes in \( n \) trials.
  • The formula for the binomial coefficient is \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \).
  • In probability, it helps to calculate the likelihood of getting a certain number of successes in a series of Bernoulli trials.
In deriving the PMF of the negative binomial distribution, the binomial coefficient is handy for figuring out the probability of getting a specific number of successes before a specified trial.
Random Variable
A random variable is a mathematical function that assigns numerical outcomes to the possible events in a probability space. They are crucial in probability and statistics, used to quantify uncertainties.
  • Random variables can be discrete or continuous. Discrete random variables, like in our scenario, have a countable number of possible values.
  • They help in modeling real-world occurrences like the total number of trials needed for a certain number of successes.
In the exercise presented, the random variable \( Y \) is defined to understand how many trials are required to achieve a particular number of successes \( r \).
The usage of random variables significantly simplifies the handling of probabilities by allowing complex scenarios to be broken down into more manageable numerical forms.

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Most popular questions from this chapter

. Suppose that \(20 \%\) of all individuals have an adverse reaction to a particular drug. A medical researcher will administer the drug to one individual after another until the first adverse reaction occurs. Define an appropriate random variable and use its distribution to answer the following questions. a. What is the probability that when the experiment terminates, four individuals have not had adverse reactions? b. What is the probability that the drug is administered to exactly five individuals? c. What is the probability that at most four individuals do not have an adverse reaction? d. How many individuals would you expect to not have an adverse reaction, and to how many individuals would you expect the drug to be given? e. What is the probability that the number of individuals given the drug is within 1 standard deviation of what you expect?

A student who is trying to write a paper for a course has a choice of two topics, \(A\) and \(B\). If topic \(A\) is chosen, the student will order two books through interlibrary loan, whereas if topic \(\mathrm{B}\) is chosen, the student will order four books. The student believes that a good paper necessitates receiving and using at least half the books ordered for either topic chosen. If the probability that a book ordered through interlibrary loan actually arrives in time is \(.9\) and books arrive independently of one another, which topic should the student choose to maximize the probability of writing a good paper? What if the arrival probability is only .5 instead of .9?

Twenty pairs of individuals playing in a bridge toumament have been seeded \(1, \ldots, 20\). In the first part of the toumament, the 20 are randomly divided into 10 east-west pairs and 10 northsouth pairs. a. What is the probability that \(x\) of the top 10 pairs end up playing east- west? b. What is the probability that all of the top five pairs end up playing the same direction? c. If there are \(2 n\) pairs, what is the pmf of \(X=\) the number among the top \(n\) pairs who end up playing east-west? What are \(E(X)\) and \(V(X)\) ?

Of the people passing through an airport metal detector, \(.5 \%\) activate it; let \(X=\) the number among a randomly selected group of 500 who activate the detector. a. What is the (approximate) pme of \(X\) ? b. Compute \(P(X=5)\). c. Compute \(P(5 \leq X)\).

A concrete beam may fail either by shear \((S)\) or flexure \((F)\). Suppose that three failed beams are randomly selected and the type of failure is determined for each one. Let \(X=\) the number of beams among the three selected that failed by shear. List each outcome in the sample space along with the associated value of \(X\).
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