Defining exponential distribution in R to estimate

Question

I have a bunch of random variables (X1,....,Xn) which are i.i.d. Exp(1/2) and represent the duration of time of a certain event. So this distribution has obviously an expected value of 2, but I am having problems defining it in R. I did some research and found something about a so-called Monte-Carlo Stimulation, but I don't seem to find what I am looking for in it.

An example of what i want to estimate is: let's say we have 10 random variables (X1,..,X10) distributed as above, and we want to determine for example the probability P([X1+...+X10<=25]).

Thanks.

Answer 1

You don't actually need monte carlo simulation in this case because:

If Xi ~ Exp(λ) then the sum (X1 + ... + Xk) ~ Erlang(k, λ) which is just a Gamma(k, 1/λ) (in (k, θ) parametrization) or Gamma(k, λ) (in (α,β) parametrization) with an integer shape parameter k.

From wikipedia (https://en.wikipedia.org/wiki/Exponential_distribution#Related_distributions)

So, P([X1+...+X10<=25]) can be computed by

pgamma(25, shape=10, rate=0.5)     

Answer 2

Are you aware of rexp() function in R? Have a look at documentation page by typing ?rexp in R console.

A quick answer to your Monte Carlo estimation of desired probability:

mean(rowSums(matrix(rexp(1000 * 10, rate = 0.5), 1000, 10)) <= 25)

I have generated 1000 set of 10 exponential samples, putting them into a 1000 * 10 matrix. We take row sum and get a vector of 1000 entries. The proportion of values between 0 and 25 is an empirical estimate of the desired probability.

Thanks, this was helpful! Can I use replicate with this code, to make it look like this: F <- function(n, B=1000) mean(replicate(B,(rexp(10, rate = 0.5)))) but I am unable to output the right result.

replicate here generates a matrix, too, but it is an 10 * 1000 matrix (as opposed to a 1000* 10 one in my answer), so you now need to take colSums. Also, where did you put n?

The correct function would be

F <- function(n, B=1000) mean(colSums(replicate(B, rexp(10, rate = 0.5))) <= n)

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For non-Monte Carlo method to your given example, see the other answer. Exponential distribution is a special case of gamma distribution and the latter has additivity property.

I am giving you Monte Carlo method because you name it in your question, and it is applicable beyond your example.