1. MSV 30: A Close Approximation
MSV 30: A Close Approximation

2. Poisson distribution Po(n × 0.01).
Homer is using a
Binomial distribution to model
a random variable X.
He says
X ~ B(n, 0.01),
where n is large.
He approximates this with the
Poisson distribution Po(n × 0.01).

3. P(X = 2) calculated using B(n, 0.01)
Homer finds that
P(X = 2) calculated using B(n, 0.01)
and P(X = 2) calculated using P(n × 0.01)
are extremely close.
In fact, for no value of n
could they be closer.
What is n in this case?

4. If X ~ P(n × 0.01), then P(X = 2) = e-0.01n(0.01n)2/2.
Answer
If X ~ B(n, 0.01), then
P(X = 2) = (0.01)2(0.99)n-2n(n-1)/2.
If X ~ P(n × 0.01), then P(X = 2) = e-0.01n(0.01n)2/2.
Plotting y = (0.01)2(0.99)x-2x(x-1)/2 - e-0.01x(0.01x)2/2 gives this:

5. There are two possible points at which the curve and the x-axis cross.
Zooming in shows that these points
are close to x = 59 and x = 341.
The value of y at 59 is 3.51 × 10-6, and y at 341 is -1.37× 10-7.
So since ‘for no other value of n could they be closer,’
X ~ B(341, 0.01) ≈ P(3.41).
For P(X=2), the approximation is 2.63 × 10-6 % out.

6. So approximation is 0.71% out here.
For X ~ B(341, 0.01), P(X = 1) = …
For X ~ P(3.41), P(X = 1) =
So approximation is 0.71% out here.

7. is written by Jonny Griffiths
With thanks to the Simpson family
is written by Jonny Griffiths