1. ## Poisson Distribution HELP!!!

Hi, i have an exam on monday and am really confused as i dont know how on earth to answer questions on poisson approximation / distribution . Heres an example question ive found which i cant do. If anyone can give me a hand i would be so grateful!!

" Suppose that a random variable X has a Poisson Distribution with parameter λ. Show that it's Probability Generating Function is given by Gx(S) = e^-λ+λs. Using this or otherwise, obtain the expected value of X."

Another question is

"Suppose that on average, 1% of a certain brand of christmas lightbulbs are defective. Compute the probability that in a box of 25 lightbulbs, there will at most be one defective bulb. Use the Poisson Approximation to compute the same probability and briefly explain whether a close match could be anticipated (Answers to 4 d.p)"

2. Originally Posted by Bexii
Hi, i have an exam on monday and am really confused as i dont know how on earth to answer questions on poisson approximation / distribution . Heres an example question ive found which i cant do. If anyone can give me a hand i would be so grateful!!

" Suppose that a random variable X has a Poisson Distribution with parameter λ. Show that it's Probability Generating Function is given by Gx(S) = e^-λ+λs. Using this or otherwise, obtain the expected value of X."

[snip]
Definition: $G_X(s) = E\left(s^X\right) = \sum_{0}^{+ \infty} s^x \frac{e^{-\lambda} \lambda^x}{x!} = e^{-\lambda} \sum_{0}^{+ \infty}\frac{(s \lambda)^x}{x!}$

and the sum is recognised as the Maclaurin series for $e^{s \lambda} \, ....$

Definition: $E(X) = \left. \frac{dG}{ds}\right|_{s = 1} = \, ....$

3. Originally Posted by Bexii
[snip]
"Suppose that on average, 1% of a certain brand of christmas lightbulbs are defective. Compute the probability that in a box of 25 lightbulbs, there will at most be one defective bulb. Use the Poisson Approximation to compute the same probability and briefly explain whether a close match could be anticipated (Answers to 4 d.p)"

Let X be the random variable number of defective bulbs.

X ~ Binomial(n = 25, p = 0.01)

Calculate Pr(X = 0) + Pr(X = 1).

Poisson approximation:

$\lambda = np = 0.25$.

Therefore $\Pr(X = x) = \frac{e^{-0.25} (0.25)^x}{x!}$

Calculate Pr(X = 0) + Pr(X = 1).

What are the conditions for using the Poisson approximation to the binomial distribution? Are they met in this question?