I have the following integral

           (x^-0.5)                                   ;in x=[0.01,1]
To solve this using Importance Sampling MC integration, one needs to select
an importance pdf that is approximately the same as the function plot

My R code to solve the same is this :

#function 1 - importance sampling
w <- function(x) dunif(x,0.01,1)/dbeta(x,0.7,1)
f <- function(x) x^(-0.5)
X <- rbeta(1000,0.7,1)
Y <- w(X)*f(X)
c(mean(Y),var(Y))
True integral value - 1.8
Using the Importance Sampling code above - 1.82 (where my importance PDF is
Beta(0.7,1)

which is quite alright so I'm assuming the code is correct.




---------------------------------------------
However I now have two bivariate functions that look like this in intervals
[x,y] in [-pi,pi] and [x,y] in [-5,5] respectively -

[image: bivariate function 1]


Could anyone guide on how to perform MC Importance Sampling for these
functions? I know I could select two independent distributions however, I
have no idea about how to choose the functions or coding it like the way
above.
-- 

Regards and Cheers,
Pooja Voladoddi
+91-90351 93110




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