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Math Help - MGF Problem

  1. #1
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    MGF Problem

    Hi

    Can someone help with the pointing me in right direction with the following

    "Use the MGF to compute the mean and variance of the distribution of x which has the following probability distribution

    f(x) = x+1/14 if x=1,2,3,4

    Many thanks
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  2. #2
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    Quote Originally Posted by statsnewbie View Post
    Hi

    Can someone help with the pointing me in right direction with the following

    "Use the MGF to compute the mean and variance of the distribution of x which has the following probability distribution

    f(x) = x+1/14 if x=1,2,3,4

    Many thanks
    The mass density function is f(x) = \frac{x + 1}{14} where x = 1, 2, 3, 4.

    By definition, the moment generating function m(t) for a random variable X is

    E\left( e^{tX}\right) = \Pr(X = x_1) \, e^{tx_1} + \Pr(X = x_2) \, e^{tx_2} + \Pr(X = x_3) \, e^{tx_3} + .......

    So for your distribution:

    m(t) = \Pr(X = 1) \, e^{t} + \Pr(X = 2) \, e^{2 t} + \Pr(X = 3) \, e^{3 t} + \Pr(X = 4) \, e^{4 t}


     = \frac{1}{14} \left( 2 \, e^{t} + 3 \, e^{2 t} + 4 \, e^{3 t} + 5 \, e^{4 t}\right) .

    By definition: \mu = E(X) = \frac{dm}{dt} evaluated at t = 0:

    ............

    By definition: E(X^2) = \frac{d^2 m}{d t^2} evaluated at t = 0:

    .............

    Formula: Var(X) = E(X^2) - \mu^2:

    .............

    I've left the details for you to fill in.
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  3. #3
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    MGF Problem

    Thanks

    One further question.

    If t=0 does that mean that each element of (e) evaluates to zero or have I missed something.
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  4. #4
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    Quote Originally Posted by statsnewbie View Post
    Thanks

    One further question.

    If t=0 does that mean that each element of (e) evaluates to zero or have I missed something.
    ?
    You find the appropriate derivatives and then substitute t = 0. You do know that e^0 = 1 I hope .....
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  5. #5
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    MGF Problem

    Yr prompt reply is appreciated

    e^0=1. Of course............ it's what I meant but not what I wrote!!! It's still too early!!!!

    Thanks a lot!!!!
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