I wish to integrate (1/y)*(2/(1+(log(y))^2))
from 0 to 1. Wolfram alpha tells me this should be pi. But when I do monte carlo integration in R, I keep getting 3.00 and 2.99 after trying over 10 times. This is what I have done:
y=runif(10^6)
f=(1/y)*(2/(1+(log(y))^2))
mean(f)
I copied the exact function into wolfram alpha to check that the integral should be pi
I tried to check if my y is properly distributed by checking it's mean and plotting a historgram, and it seems to be ok. Could there be something wrong with my computer?
Edit: Maybe someone else could copy my code and run it themselves, to confirm that it isn't my computer acting up.
Ok, first let's start with simple transformation, log(x) -> x
, making integral
I = S 2/(1+x^2) dx, x in [0...infinity]
where S
is integration sign.
So function 1/(1+x^2) is falling monotonically and reasonable fast. We need some reasonable PDF to sample points in [0...infinity] interval, such that most of the region where original function is significant is covered. We will use exponential distribution with some free parameter which we will use to optimize sampling.
I = S 2/(1+x^2)*exp(k*x)/k k*exp(-k*x) dx, x in [0...infinity]
So, we have k*e-kx as properly normalized PDF in the range of [0...infinity]. Function to integrate is (2/(1+x^2))*exp(k*x)/k
. We know that sampling from exponential is basically -log(U(0,1))
, so code to do that is very simple
k <- 0.05
# exponential distribution sampling from uniform vector
Fx <- function(x) {
-log(x) / k
}
# integrand
Fy <- function(x) {
( 2.0 / (1.0 + x*x) )*exp(k*x) / k
}
set.seed(12345)
n <- 10^6L
s <- runif(n)
# one could use rexp() as well instead of Fx
# x <- rexp(n, k)
x <- Fx(s)
f <- Fy(x)
q <- mean(f)
print(q)
Result is equal to 3.145954
, for seed 22345
result is equal to 3.135632
, for seed 32345
result is equal to 3.146081
.
UPDATE
Going back to original function [0...1] is quite simple
UPDATE II
changed per prof.Bolker suggestion
If you love us? You can donate to us via Paypal or buy me a coffee so we can maintain and grow! Thank you!
Donate Us With