Question

In r make a simulation to verify the following theorem: if X₁ ∼ χ²_n1 , X₂ ∼ χ ²_n2 , and X₁ and X₂ are independent, then X₁ + X₂ ∼ χ ²_{n1+n2 .}

Please provide appropriate code for this problem.

Answer 1

You can use the code below:

CODE:

check = function(n1,n2,df1,df2){
x1 = rchisq(n1,2*df1)
x2 = rchisq(n2,2*df2)
y = x1+x2
library(e1071)
probplot(y,"qchisq",df = 2*(df1+df2))
}
check(2000,2000,10,20)

The above code will create the probability plot to visualize whether x1+x2 follows the required distribution.

You can see the plots for X1+X2 are falls on the straight lines in the graph, which indicates that they follow the chi-square distribution with df 2(n1 + n2).