question about probability urgent pls?

The range of discrete random variable X is the set of all natural numbers and P(X = k)=C/9^k for k=1,2,...... Determine C and the moment generating function of X.

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  • 8 years ago
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    We know that the sum over all k of P(X=k) must be 1.

    So C (1/9 + 1/(9^2) + 1/(9^3) + .....) = 1

    This is an infinite geometric series whose sum is:

    C * (1/9) * 1/(1-(1/9)) = C * (1/9) * (9/8) = C / 8

    So C / 8 = 1, so C = 8.

    The moment generating function of X is defined as M_X(t) = E(e^tX)

    E(e^tX) = sum of e^tk * P(X=k) over all k>=1.

    = sum of e^tk * 8/9^k

    = sum of 8 * (e^t/9)^k

    This is another geometric series which only converges if |e^t / 9| < 1, i.e. if -1 < e^t / 9 < 1

    Rearranging this gives ln(1/9) < t < ln (9)

    When t is in this range, the sum of the series is 8 * e^t/9 * 1/(1-e^t/9) = 8 * e^t/9 * 9/(9 - e^t) = 8e^t / (9 - e^t).

    So M_X(t) = 8e^t / (9 - e^t) for ln(1/9) < t < ln(9).

    When t is outsie

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