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)"
Thankyou for any help you can give me, i am literally clueless and am so worried about this exam coming up!!!
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:
.
Therefore
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?