# Negative Binomial Distribution Examples And Solutions Pdf

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- Negative Binomial distribution
- Negative Binomial Distribution
- Negative binomial distribution
- Negative Binomial Distribution

## Negative Binomial distribution

In this lesson, we cover the negative binomial distribution and the geometric distribution. As we will see, the geometric distribution is a special case of the negative binomial distribution. A negative binomial experiment is a statistical experiment that has the following properties:.

Consider the following statistical experiment. You flip a coin repeatedly and count the number of times the coin lands on heads. You continue flipping the coin until it has landed 5 times on heads. This is a negative binomial experiment because:. A negative binomial random variable is the number X of repeated trials to produce r successes in a negative binomial experiment.

The probability distribution of a negative binomial random variable is called a negative binomial distribution. The negative binomial distribution is also known as the Pascal distribution. Suppose we flip a coin repeatedly and count the number of heads successes. If we continue flipping the coin until it has landed 2 times on heads, we are conducting a negative binomial experiment.

The negative binomial random variable is the number of coin flips required to achieve 2 heads. In this example, the number of coin flips is a random variable that can take on any integer value between 2 and plus infinity. The negative binomial probability distribution for this example is presented below. The negative binomial probability refers to the probability that a negative binomial experiment results in r - 1 successes after trial x - 1 and r successes after trial x.

For example, in the above table, we see that the negative binomial probability of getting the second head on the sixth flip of the coin is 0. Given x , r , and P , we can compute the negative binomial probability based on the following formula:.

Negative Binomial Formula. Suppose a negative binomial experiment consists of x trials and results in r successes.

If the probability of success on an individual trial is P , then the negative binomial probability is:. If we define the mean of the negative binomial distribution as the average number of trials required to produce r successes, then the mean is equal to:. As if statistics weren't challenging enough, the above definition is not the only definition for the negative binomial distribution.

Two common alternative definitions are:. The moral: If someone talks about a negative binomial distribution, find out how they are defining the negative binomial random variable. On this website, when we refer to the negative binomial distribution, we are talking about the definition presented earlier. That is, we are defining the negative binomial random variable as X , the total number of trials required for the binomial experiment to produce r successes.

The geometric distribution is a special case of the negative binomial distribution. It deals with the number of trials required for a single success. Thus, the geometric distribution is negative binomial distribution where the number of successes r is equal to 1. An example of a geometric distribution would be tossing a coin until it lands on heads. We might ask: What is the probability that the first head occurs on the third flip?

That probability is referred to as a geometric probability and is denoted by g x ; P. The formula for geometric probability is given below. Geometric Probability Formula. Suppose a negative binomial experiment consists of x trials and results in one success.

If the probability of success on an individual trial is P , then the geometric probability is:. The problems below show how to apply your new-found knowledge of the negative binomial distribution see Example 1 and the geometric distribution see Example 2. As you may have noticed, the negative binomial formula requires many time-consuming computations. The Negative Binomial Calculator can do this work for you - quickly, easily, and error-free.

Use the Negative Binomial Calculator to compute negative binomial probabilities. The calculator is free. It can found in the Stat Trek main menu under the Stat Tools tab. Or you can tap the button below. Example 1 Bob is a high school basketball player. That means his probability of making a free throw is 0.

During the season, what is the probability that Bob makes his third free throw on his fifth shot? Solution: This is an example of a negative binomial experiment. The probability of success P is 0. Thus, the probability that Bob will make his third successful free throw on his fifth shot is 0. Let's reconsider the above problem from Example 1. This time, we'll ask a slightly different question: What is the probability that Bob makes his first free throw on his fifth shot?

Solution: This is an example of a geometric distribution, which is a special case of a negative binomial distribution. Therefore, this problem can be solved using the negative binomial formula or the geometric formula. We demonstrate each approach below, beginning with the negative binomial formula.

We enter these values into the negative binomial formula. Alternative Views of the Negative Binomial Distribution As if statistics weren't challenging enough, the above definition is not the only definition for the negative binomial distribution.

Two common alternative definitions are: The negative binomial random variable is R , the number of successes before the binomial experiment results in k failures. Negative Binomial Calculator As you may have noticed, the negative binomial formula requires many time-consuming computations.

Negative Binomial Calculator.

## Negative Binomial Distribution

The probability density function is therefore given by. The distribution function is then given by. The negative binomial distribution is implemented in the Wolfram Language as NegativeBinomialDistribution [ r , p ]. Since ,. The raw moments are therefore. Note that Beyer , p. The mean , variance , skewness and kurtosis excess are then.

What is the probability that the first strike comes on the third well drilled? Since a geometric random variable is just a special case of a negative binomial random variable, we'll try finding the probability using the negative binomial p. It is at the second equal sign that you can see how the general negative binomial problem reduces to a geometric random variable problem. What is the mean and variance of the number of wells that must be drilled if the oil company wants to set up three producing wells? Breadcrumb Home 11

In this lesson, we cover the negative binomial distribution and the geometric distribution. As we will see, the geometric distribution is a special case of the negative binomial distribution. A negative binomial experiment is a statistical experiment that has the following properties:. Consider the following statistical experiment. You flip a coin repeatedly and count the number of times the coin lands on heads.

## Negative binomial distribution

Documentation Help Center. X , R , and P can be vectors, matrices, or multidimensional arrays that all have the same size, which is also the size of Y. A scalar input for X , R , or P is expanded to a constant array with the same dimensions as the other inputs. Note that the density function is zero unless the values in X are integers. The simplest motivation for the negative binomial is the case of successive random trials, each having a constant probability P of success.

It refers to the probabilities associated with the number of successes in a hypergeometric experiment. An introduction to the hypergeometric distribution. The hypergeometric distribution differs from the binomial distribution in the lack of replacements. Said another way, a discrete random variable has to be a whole, or counting, number only. The three discrete distributions we discuss in this article are the binomial distribution, hypergeometric distribution, and poisson distribution.

### Negative Binomial Distribution

In probability theory and statistics , the negative binomial distribution is a discrete probability distribution that models the number of successes in a sequence of independent and identically distributed Bernoulli trials before a specified non-random number of failures denoted r occurs. In such a case, the probability distribution of the number of non-6s that appear will be a negative binomial distribution. We could just as easily say that the negative binomial distribution is the distribution of the number of failures before r successes. When applied to real-world problems, outcomes of success and failure may or may not be outcomes we ordinarily view as good and bad, respectively. This article is inconsistent in its use of these terms, so the reader should be careful to identify which outcome can vary in number of occurrences and which outcome stops the sequence of trials.

The Negative Binomial distribution estimates the number of failures there will be before s successes are achieved where there is a probability p of success with each trial. Examples of the Negative Binomial distribution are shown below:. The first use is when we know that we will stop at the s th success. The second is when we only know that there had been a certain number of successes. For example, a hospital has received a total of 17 people with a rare disease in the last month.

Negative Binomial distribution: number of trials until the rth success Negative Binomial. Distribution. Example 37 cont'd. Solution. 1.

#### Basic Theory

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In particular, it follows from part a that any event that can be expressed in terms of the negative binomial variables can also be expressed in terms of the binomial variables.