Negative Binomial

FOR SOLVED PREVIOUS PAPERS OF ISS KINDLY CONTACT US ON OUR WHATSAPP NUMBER 9009368238

FOR SOLVED PREVIOUS PAPERS OF ISS KINDLY CONTACT US ON OUR WHATSAPP NUMBER 9009368238

---

---

Negative Binomial

The Negative Binomial distribution is a versatile probability distribution that models the number of trials needed to achieve a specified number of successes in a sequence of independent Bernoulli trials. It generalizes the Geometric distribution and is widely used in fields like statistics, biology, and engineering. In this blog, we’ll explore the Negative Binomial distribution in detail, including its definition, properties, and real-world applications.

---

1. What is the Negative Binomial Distribution?

The Negative Binomial distribution models the number of trials required to achieve a fixed number of successes in a sequence of independent Bernoulli trials, each with the same probability of success ( p ).

Definition:

A random variable ( X ) follows a Negative Binomial distribution with parameters ( r ) (number of successes) and ( p ) (probability of success in each trial) if its probability mass function (PMF) is:
[
P(X = k) = \binom{k – 1}{r – 1} p^r (1 – p)^{k – r} \quad \text{for } k = r, r + 1, r + 2, \dots
]
Here:

• ( k ) is the total number of trials.
• ( r ) is the number of successes.
• ( p ) is the probability of success in each trial.
• ( \binom{k – 1}{r – 1} ) is the binomial coefficient, representing the number of ways to arrange ( r – 1 ) successes in the first ( k – 1 ) trials.

---

2. Properties of the Negative Binomial Distribution

1. Mean (Expected Value):
   [
   E[X] = \frac{r}{p}
   ]
   This represents the average number of trials needed to achieve ( r ) successes.
2. Variance:
   [
   \text{Var}(X) = \frac{r (1 – p)}{p^2}
   ]
   This measures the spread of the distribution.
3. Special Case: Geometric Distribution:

• When ( r = 1 ), the Negative Binomial distribution reduces to the Geometric distribution, which models the number of trials until the first success.

1. Shape:

• The distribution is right-skewed, with a longer tail for smaller values of ( p ).

---

3. Example: Applying the Negative Binomial Distribution

Let’s walk through an example to see how the Negative Binomial distribution works in practice.

Problem:

Suppose you’re conducting a series of independent Bernoulli trials with a success probability of ( p = 0.3 ). What is the probability that you’ll need exactly 10 trials to achieve 3 successes?

Solution:

1. Define the Parameters:

• ( r = 3 ) (number of successes).
• ( p = 0.3 ) (probability of success in each trial).
• ( k = 10 ) (total number of trials).

1. Compute the Probability:

• Using the Negative Binomial PMF:
  [
  P(X = 10) = \binom{10 – 1}{3 – 1} (0.3)^3 (0.7)^{10 – 3}
  ]
  Calculate the binomial coefficient:
  [
  \binom{9}{2} = \frac{9!}{2!(9 – 2)!} = 36
  ]
  Now, plug the values into the PMF:
  [
  P(X = 10) = 36 \cdot (0.3)^3 \cdot (0.7)^7 \approx 36 \cdot 0.027 \cdot 0.0824 \approx 0.0797
  ]

1. Conclusion:
   The probability of needing exactly 10 trials to achieve 3 successes is approximately 7.97%.

---

4. Applications of the Negative Binomial Distribution

The Negative Binomial distribution is widely used in various fields to model scenarios involving the number of trials until a specified number of successes. Here are some examples:

a. Quality Control:

• Example: Number of items inspected until a certain number of defects are found.
• If the probability of a defect is ( p = 0.05 ), the distribution can model the number of items inspected until 5 defects are found.

b. Biology:

• Example: Number of trials until a certain number of mutations occur.
• If the probability of a mutation is ( p = 0.01 ), the distribution can model the number of trials until 3 mutations occur.

c. Marketing:

• Example: Number of customer interactions until a certain number of sales are made.
• If the probability of a sale is ( p = 0.1 ), the distribution can model the number of interactions until 2 sales are made.

d. Sports:

• Example: Number of games played until a certain number of wins are achieved.
• If the probability of winning a game is ( p = 0.4 ), the distribution can model the number of games played until 4 wins are achieved.

---

5. Key Takeaways

• The Negative Binomial distribution models the number of trials needed to achieve a specified number of successes in a sequence of independent Bernoulli trials.
• It is characterized by two parameters: ( r ) (number of successes) and ( p ) (probability of success in each trial).
• The mean of the distribution is ( \frac{r}{p} ), and the variance is ( \frac{r (1 – p)}{p^2} ).
• It is widely used in quality control, biology, marketing, and sports.

---

6. Why Does the Negative Binomial Distribution Matter?

The Negative Binomial distribution is a powerful tool for modeling and analyzing scenarios involving the number of trials until a specified number of successes. By understanding it, you can:

• Predict the number of trials needed to achieve a certain number of successes.
• Make informed decisions in fields like quality control, biology, and marketing.
• Build accurate models for real-world phenomena involving repeated trials.

---

Conclusion

The Negative Binomial distribution is a fundamental concept in probability and statistics, offering a way to model the number of trials needed to achieve a specified number of successes. Whether you’re inspecting products, studying mutations, or analyzing customer interactions, this distribution provides the mathematical framework to analyze and predict outcomes. By mastering the Negative Binomial distribution, you’ll be well-equipped to tackle a wide range of problems in science, engineering, and beyond.

---