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Math Help - MGF

  1. #1
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    MGF

    For the random variable:
    P(Y=0)= 1/8, P(Y=1)=1/2, P(Y=3)=3/8
    Find the MGF of Y, and E[Y] and Var(Y).

    Helpppp, I reallly don't have a clue how to do this!
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  2. #2
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    Quote Originally Posted by georgiahelo View Post
    For the random variable:
    P(Y=0)= 1/8, P(Y=1)=1/2, P(Y=3)=3/8
    Find the MGF of Y, and E[Y] and Var(Y).

    Helpppp, I reallly don't have a clue how to do this!
    If X is a finite random variable taking on values x1, x2, . . ., xn, the mean or expected value of X, written μ, or E(X), is
    μ = E(X) = x1.P(X = x1) + x2.P(X = x2) + . . . + xn.P(X = xn) = ∑ (xi.P(X = xi))If X is a random variable, its variance is defined to be
    σ2 = E( [X - μ]2 ). Its standard deviation is defined to be the square root σ of the variance.

    replace X by Y ...
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  3. #3
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    Quote Originally Posted by georgiahelo View Post
    For the random variable:
    P(Y=0)= 1/8, P(Y=1)=1/2, P(Y=3)=3/8
    Find the MGF of Y, and E[Y] and Var(Y).

    Helpppp, I reallly don't have a clue how to do this!
    Start by applying the defintions!

    By definition of MGF: m(t) = E(e^{tY}) = \frac{1}{8} + \frac{1}{2} e^t + \frac{3}{8} e^{3t}.

    By definition: E(Y) = \left. \frac{dm}{dt}\right|_{t = 0} = ....

    By definition: Var(Y) = E(Y^2) - [E(Y)]^2. And by definition E(Y^2) = \left.\frac{d^2m}{dt^2}\right|_{t = 0} = ....
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  4. #4
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    MGF's

    Thankyou, I think I've done it now!
    I have another question!
    If Y1 and Y2 are independent random variables with the same PMF as the original posted question, how do I find the MGF of W=Y1+Y2?
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  5. #5
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    Quote Originally Posted by georgiahelo View Post
    Thankyou, I think I've done it now!
    I have another question!
    If Y1 and Y2 are independent random variables with the same PMF as the original posted question, how do I find the MGF of W=Y1+Y2?
    You're expected to know the theorem given here regarding the moment generating function of a sum of independent random variables: Moment-generating function - Wikipedia, the free encyclopedia


    Edit: Asked here: http://www.mathhelpforum.com/math-he...ution-mgf.html. Thread closed.
    Last edited by mr fantastic; December 9th 2009 at 06:30 PM.
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