# Negative Binomial Distribution

**Episode #8 of the course Theory of probability by Polina Durneva**

Good morning!

Today, we will complete our discussion on discrete probability distributions by talking about negative binomial distribution. For this distribution, you need to run a certain number of trials (n) until you get the wanted number of successes (r). To some extent, this distribution is the inverse of traditional binomial distribution, but in this case, you are interested in the number of failures, not successes.

**What Is Negative Binomial Distribution?**

Each trial of negative binomial distribution follows Bernoulli distribution. It means that each trial can either be a success (normally denoted as 1) or a failure (denoted as 0). To better understand the main idea behind this distribution, let’s proceed to an example. Let’s assume that you are in the room with some strangers and you want to find 3 people who have been abroad more than once in their lives. In this case, finding a stranger who has been abroad more than once is a success. Assume that you want to know the probability of getting these 3 successes after asking 10 people. The solution to this problem will be based on negative binomial distribution! The wanted number of success is 3, and the total number of trials is 10.

Before we solve the problem using the formula, we also need to discuss a series of assumptions for negative binomial distribution:

1. The outcome for each trial is either a success or a failure.

2. The probability of success and failure are consistent from one trial to another.

3. Each trial is independent from each other.

4. The trials keep going on until a given number of successes occur.

**Formula for Negative Binomial Distribution**

The formula for Negative Binomial Distribution takes the following form: P(x) = _{x-1}C_{r-1} * p^{r} * (1 – p)^{x-r}, where p is the probability of getting a success, x is the number of trials, and r is the given number of successes. _{x-1}C_{r-1} stands for the number of all possible ways of getting r – 1 objects from a set of x – 1 objects. To better understand the formula, let’s use our example from the previous section.

We want 3 successes with probabilities of occurring of 0.5. To find the probability that 10 trials will be enough, let’s use the formula: _{10-1}C_{3-1} * 0.5^{3} * (1 – 0.5)^{10-3} = 0.035 = 3.5%. This probability is too low because we probably need fewer trials to get 3 successes (see section below).

**Expected Value of Negative Binomial Distribution**

The expected value of negative binomial distribution takes time to prove, so let’s just proceed to the formula: r/p. In our previous example, r is 3 and p is 0.5. It means that the expected value would be: 3/0.5 = 6. It means that on average, we would need 6 trials to get 3 successes.

**Variance of Negative Binomial Distribution**

The variance of negative binomial distribution can be calculated using the following formula: r * (1 – p) / p^{2}. In our previous example, the variance would be: 3 * (0.5) / (0.5)^{2} = 6. (Please note that in this example, the variance is equal to the expected value. This, however, does not happen all the time.)

That’s it for today! Tomorrow, we will start discussing our first continuous probability distribution: exponential distribution.

See you soon,

Polina

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