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Math Help - Poisson Distribution HELP!!!

  1. #1
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    Exclamation 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)"

    Thankyou for any help you can give me, i am literally clueless and am so worried about this exam coming up!!!
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  2. #2
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    Quote Originally Posted by Bexii View Post
    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} = \, ....
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    Quote Originally Posted by Bexii View Post
    [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)"

    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:

    \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?
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