Question

The number of particles emitted by a radioactive source during a ten second interval has
the Poisson distribution with mean λ. The radioactive source is observed over twenty non-
overlapping intervals of ten seconds each. The number of particles emitted during each
interval are: 4, 1, 3, 1, 3, 3, 3, 1, 1, 1, 6, 4, 4, 2, 2, 1, 1, 4, 2, 5. Use R and show R code!
a. Using method of moments (show your work) find an estimate for λ.
b. Create a function to represent your likelihood function from a Poisson
distribution and use the sample given above. Plot the function for a range
of appropriate possible values of λ. Find the value of λ that maximized the
likelihood.
c. Check if the theoretical maximum likelihood estimator for λ agrees with
the plot from part (b). Explain your results/observations.

Answer 1

a.

R code:

x=c(4, 1, 3, 1, 3, 3, 3, 1, 1, 1, 6, 4, 4, 2, 2, 1, 1, 4, 2, 5)
m=mean(x)
lambda=seq(0,4,by=0.01)
n1=length(x)
n2=length(lambda)
p=matrix(0,n1,n2)
p1=1:n2*0
for(i in 1:n1)
for(j in 1:n2)
{
{
p[i,j]=exp(-lambda[j])*lambda[j]^x[i]/factorial(x[i])
}
}
for(j in 1:n2)
{
p1[j]=prod(p[,j])
}
p1
plot(lambda,p1,lwd=2,type="l",xlab=expression(lambda),ylab=expression(L(lambda)))
abline(v=mean(x),lwd=2,col=2)

m

Output:

m= 2.6

From the above plot it is obseved that at