Statistics 430
Assignment
Due April 3
This homework assignment considers another use of the central limit theorem. First, it will lead you
through the derivation, one that is quite similar to the one done in class. Second, you’ll use the resulting
normal approximation.
Throughout this assignment, the random variable Xdenotes a Poisson random variable with parameter
λand PMF given by
p(x) = e−λλx
x!, x = 0,1, . . .
We’ll consider what happens to this density as λ→ ∞. To keep the algebra manageable, assume that λis
an integer.
1. Show (for integer valued λ > 0) that the peak (mode) of the Poisson density is at λ−1 and at λ. That
is, show that p(λ−1) = p(λ) and that the common value is the maximum of p(x). (Hint: Look at the
ratio of p(x)/p(x−1) for xnear λ.)
2. Show that we can write the ratio of probability at λ+x(for x > 0) to the probability at the mode
(which is also the mean) λas the following sum:
p(λ+x)
p(λ)=p(λ+x)
p(λ+x−1)
p(λ+x−1)
p(λ+x−2) ··· p(λ+ 1)
p(λ).(1)
3. Show that a typical ratio in the product (1) has the form
p(j)
p(j−1) =λ
j.
4. Combine the results of the previous two steps to show that the log of the big product in (1) can be
written as the sum
log p(λ+x)
p(λ)=−
x
X
j=1
log(1 + j/λ).(2)
5. Approximate the sum in (2) using log(1 + )≈for “close” to zero. Also approximate the sum of
the first xintegers as Px
j=1 j≈x2/2, arriving at
log p(λ+x)
p(λ)≈ − x2
2λ.(3)
6. Because of the use of the approximation to the log in the prior step, under what conditions (i.e., for
what values of x) will the approximation in equation (3) be useful?
7. A very famous approximation for the factorial finishes up the derivation. The approximation is known
as Stirling’s formula, given by (for positive integers)
j!≈jje−jp2πj . (4)
Compare Stirling’s approximation for j! to the exact value for j= 5,10,20, and 50. How accurate is
the approximation?
