How to plot a factorial function in R

Question

I'm trying to plot:

Using the following R code with no success:

N= seq(from=150, to=2000)
P=((factorial(60) / factorial(50))*(factorial(N-60) /factorial(N-150))) /(factorial(N) /factorial(N-100))
plot(N,P)

Answer 1

Almost always, probability expression involving factorial is some result of "N choose K" computation:

But it is very inefficient to compute this via factorial, and most importantly, it is not numerically stable. Have a look at your code using factorial() : you got NaN .

In R, the choose(N, K) function computes "N choose K" fast and stably.

Now, a careful inspection of your given formulation shows that it is equivalent to:

choose(N-100, 50) / choose(N, 60)

So, you can do:

P <- choose(N-100, 50) / choose(N, 60)
plot(N, P, type = "l")

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Follow-up

Hi, this is a very efficient function. But mean, mode, and median of this plot doesn't match the ones I have in my course materials for the same plot? The mean should be 727, Mode= 600, median= 679!! How can I get these descriptives from your suggested plot?

I am confused by what your course material is trying to do. The probability you give is conditional probability P(D | N) , ie, a probability for random variable D . While we sketch P against N . Hence, the plot above is not a probability mass function! Then, how can we use it to compute statistics like mean, mode and median, for random variable N ???

Well anyway, since you ask and insist on getting an answer, let's pretend this is a probability mass function for random variable N . But since it is not a true one, sum(P) is not or even close to 1. We actually have sum(P) = 3.843678e-12 . So, to use it as a proper probability mass function, we need to normalize it first.

P <- P / sum(P)

Now P sum up to 1.

To compute mean, we do

sum(N * P)
# [1] 726.978

To compute mode, we do

N[which.max(P)]
# 599

To compute median, we do

N[which(cumsum(P) > 0.5)[1]]
# 679