Question

Give a method for simulating a negative binomial random variable.

Short answer

To simulate a negative binomial random variable, follow these steps: 1. Define the parameters: probability of success \(p\) (0 < p ≤ 1) and target number of successes \(r\). Initialize a counter for the number of failures \(F\) and successes \(S\) to 0. 2. Generate a random number \(U\) uniformly distributed between 0 and 1, using a built-in function (e.g., `random.random()` in Python). 3. Compare \(U\) to \(p\). If \(U ≤ p\), increment successes (S = S + 1). If \(U > p\), increment failures (F = F + 1). 4. Check if the target number of successes \(r\) has been reached. If yes, the negative binomial random variable is equal to the number of failures \(F\). 5. If the target number of successes is not reached, go back to Step 2 and repeat until the target number of successes is achieved.

Step-by-step solution

1. Define parameters and necessary variables

First, we need to define the two parameters of the negative binomial distribution: the probability of success \(p\) (0 < p ≤ 1) and the target number of successes \(r\). Additionally, initialize a counter for the number of failures \(F\) and the number of successes \(S\) to 0.

2. Generate a random number

Now we will generate a random number \(U\) uniformly distributed between 0 and 1. Many programming languages have built-in functions that can be used for this purpose (e.g., Python uses `random.random()`).

3. Compare the generated random number to the probability of success

Next, we will compare the randomly generated number \(U\) to the probability of success \(p\). If \(U\) is less than or equal to \(p\), it means we have achieved a success, so we will increment the number of successes \(S\) by 1 (S = S + 1). If \(U > p\), it means we have encountered a failure, so we will increment the number of failures \(F\) by 1 (F = F + 1).

4. Check if the target number of successes has been reached

After updating the number of successes or failures in the previous step, we will check if the target number of successes \(r\) has been reached. If yes, we will stop the simulation, and the negative binomial random variable will be equal to the number of failures, \(F\).

5. If the target number is not reached, go back to Step 2

If the target number of successes has not been reached, we will go back to Step 2 and continue the simulation by generating a new random number, comparing it to the probability of success, and updating the number of successes and failures accordingly. This process will be repeated until the target number of successes is achieved.