Question

Generate 100 Poisson (λ = 2) random numbers using the Inverse transformation method, and then compare with the theoretical mean and variance.

please let me know the explanaiton with detail, and r code, If not, at least python

Answer 1

Run below code rmin R:

(list=ls())


#### Lets find CDF i.e. F(X=x) for Poisson(m=2)
# Evaluating till X = 15 as P(X=x)is negligible after X=15
F_x=ppois(0:15,lambda = 2)
F_x


### Lets genarate 100 random numbers from U(0,1)
u=runif(100,min = 0,max = 1)

### Now create Poisson r.n. using its cdf and r.n. from U(0,1)
### If u belongs to ( F(x-1), F(x) ) then Poisson r.v. will be equal to x
X=c()
for(i in 1:100){
X=c(X,min(which(F_x>u[i])))
}

X
cat('Sample mean=', mean(X),'And theoretical mean=2')
cat('Sample variance=',var(X), 'And theoretical variance=2')

You will get below output

> rm(list=ls ) Values num [1:16] 0.135 0.40G 100L num (1:100] 0.317 0.46 int (1:100] 2 3 2 2 2 > #### Lets find CDF i.e. F(X=X) for Poisson(m=2) > # Evaluating till X = 15 as P(X=x)is negligible after X=15 > F_X=ppois(0:15, lambda = 2) > F_x [1] 0.1353353 0.4060058 0.6766764 0.8571235 0.9473470 0.9834364 0.9954662 [8] 0.9989033 0.9997626 0.9999535 0.9999917 0.9999986 0.9999998 1.0000000 [15] 1.0000000 1.0000000 > ### Lets genarate 100 random numbers from U(0,1) > urrunif(100 min = 0,max = 1) annnnnnn > ### Now create poisson r.n. using its cdf and r. n. from U(0.1) > ### If u belongs to ( F(x-1). F(x)) then poisson r.v. will be equal to x > X=C > for(i in 1:100){ + X=c(X,min(which(F_x>u[i]))) > X [1] 2 3 2 2 2 2 4 2 2 1 1 1 3 5 3 3 6 2 5 4 3 1 2 7 1 2 2 4 2 1 3 2 3 2 3 3 [37] 3 2 1 3 1 3 3 4 1 2. 2 3 4 2 1 4 4 2 3 2 2 3 2 5 1 1 5 4 2 3 3 2 3 4 2 6 [73] 4 1 2 5 2 3 2 5 3 2 2 2 1 3 6 3 3 3 3 1 4 5 3 5 2 5 6 2 > cat('Sample mean=', mean(X). 'And theoretical mean=2 \n') Sample mean= 2.82 And theoretical mean=2 > cat('Sample variance='.var(x). 'And theoretical variance=2') Sample variance= 1.906667 And theoretical variance=2