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!!!