# What is the probability of given data with the Negative binomial distribution?

A university exercise Statistics Learning and data analysis.

This is the problem:

Given that $$X=x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8,x_9, x_{10} = \left(2,3,7,8,2,4,7,5,5,7\right)$$

1. What is the mean $${\overline{x}}$$ ?

Answer: $${\overline{x}}=\frac{\left(2+3+7+8+2+4+7+5+5+7\right)}{10}=5$$

1. What is the probability of $$P\left(X\le\overline{x}\right)$$? with help from $$x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8,x_9, x_{10}$$ from a $$NegBin\left(2,\frac{1}{3}\right)$$ ?

My answer (That is wrong would be): $$P\left(X\le\overline{x}\right)=\sum_{i=0}^5\left(\frac{1}{3}\right)^{^2}\left(\frac{2}{3}\right)^i\binom{i+2-1}{i}=\sum_{i=0}^5\left(\frac{1}{3}\right)^{^2}\left(\frac{2}{3}\right)^i\binom{i+1}{i}=\sum_{i=0}^5\left(\frac{1}{3}\right)^{^2}\left(\frac{2}{3}\right)^{^i}\left(i+1\right)=0.73663$$

In the answer they stated that they drew $$x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8,x_9, x_{10}$$ 100 times and got the answer $$P\left(X\le5\right) = 0.19$$

My questions are:

1. How do you come to the same conclusion as the "right answer"?
2. Is there a different way of solving this problem?
3. How would you answer the problem (2)?

Thanks!

Update:

The data X are the number of times something is observed in a day. Meaning there are 10 days and therefore 10 observed data in X.

NegBin(r,p) = $$NegBin(r,p): fx(x) = (p)^{r}(p-1)^i\binom{i+r-1}{i}, E[X] = r(1 − p)/p$$

• There seems to be something missing. In particular the meaning of $X$ seems confusing: is it a set/multiset/vector of observations? A mean? A random variable drawn from the multiset? The mean of a bootstrap sample from the multiset? The mean of a random variable from a negative binomial distribution? Commented Sep 14, 2022 at 15:02
• It will help us if you will state the definition of the negative binomial distribution given in your text--specifically, what is the pmf. Different texts have different definitions. Commented Sep 14, 2022 at 15:05
• The problem has been updated. Appreciated the feedback. Commented Sep 15, 2022 at 8:31
• Maybe there is a way of estimating what the inverse function would be: After that check what the numbers (u) would be if X<= 5 Take the f^-1(X=5) = u Make 100 random variables from U(0,1) Check how many of these 100 random variables from U(0,1) that are less or equal to u and then draw a conclusion from this? Is this a way of solving this problem or am I thinking something irrational? Commented Sep 15, 2022 at 8:48

1. Make $$100$$ random variables between $$0$$ and $$1$$.
2. Check for every number of $$i$$ in $$\text{NegBin}(r,p)$$, where $$u > \text{NegBin}(r,p)$$, and for every single $$i$$ that you check, you will add the previous PDF function $$(i-1)$$.
3. Whenever $$u < \,($$sum of all $$\text{NegBin}(r,p)$$ for an $$i)$$, this $$i$$ will be saved as a simulation of $$\text{NegBin}(r,p)$$.
4. Then you check $$P(\text{simulations} \leq 5)$$, and this will give you the answer.