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Math Help - Density function

  1. #1
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    Density function

    Given: Suppose that a R.V Y has a probability density function by:

    f(y) = k * (y^3) * e^(-y/2) when y > 0

    f(y) = 0 elsewhere

    a) Find the value of k that makes f(y) a density function:

    b) What are the mean and standard deviation of Y?

    Sol.

    For part a) I started out the same way as in previous similar problem.
    I integrate f(y)=k*(y^3) * e^(-y/2) from 0 to infinity and make it equal to 1. I get this:

    -2 k (48 + x (24 + x (6 + x)))
    ------------------------------
    e^x/2

    If we solve it from 0 to infinity it comes out to be 0... I checked the integral with a computer program, it looks right. I wonder maybe it should be from 1 to infinity? Then it is possible to find k.
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  2. #2
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    I think I found the problem, when we integrate from 0 to infinity we get
    2*k*48 = 1
    so k= 1/96,
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by somestudent2 View Post
    Given: Suppose that a R.V Y has a probability density function by:

    f(y) = k * (y^3) * e^(-y/2) when y > 0

    f(y) = 0 elsewhere

    a) Find the value of k that makes f(y) a density function:

    b) What are the mean and standard deviation of Y?

    Sol.

    For part a) I started out the same way as in previous similar problem.
    I integrate f(y)=k*(y^3) * e^(-y/2) from 0 to infinity and make it equal to 1. I get this:

    -2 k (48 + x (24 + x (6 + x)))
    ------------------------------
    e^x/2

    If we solve it from 0 to infinity it comes out to be 0... I checked the integral with a computer program, it looks right. I wonder maybe it should be from 1 to infinity? Then it is possible to find k.
    The expression evaluates to -96k at x=0, and 0 at infinity, so your integral is 96k.

    RonL
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  4. #4
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    cool, thank you sir. I am happy that at least some of my answers match up.
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