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Thread: Integral calculus

  1. #1
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    Integral calculus

    Hi I'm Heather and I'm doing my Masters in Statistics.
    I have a calculus prob.
    I have this question about pdf:
    Suppose X be a continuous random variable with the pdf:
    f(x) = {k (2 +x^2 -x) -1 <= x <= 2
    0 otherwise
    I need to find the value of k, which is a positive constant and then obtain the cumulative distribution function (F (x)) of X.
    Can someone help please?

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  2. #2
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    Re: Integral calculus

    $\displaystyle \int_{-\infty}^{\infty}~f(x) ~dx = 1$

    that let's you evaluate $k$ and thus get $f(x)$, then

    $F(x) = \displaystyle \int_{-\infty}^x~f(u)~du$
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  3. #3
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    Re: Integral calculus

    Quote Originally Posted by romsek View Post
    $\displaystyle \int_{-\infty}^{\infty}~f(x) ~dx = 1$

    that let's you evaluate $k$ and thus get $f(x)$, then

    $F(x) = \displaystyle \int_{-\infty}^x~f(u)~du$
    So I have tried to work this out and I believe k =3/8. Is that right?
    Do I then have to work out 3/8 (2+x^2-x)? Or is that the pdf?
    Thanks.

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  4. #4
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    Re: Integral calculus

    is $f(x) = k(x^2 - x + 2),~x \in [-1,2]$ ?

    If so I get $k=\dfrac {2}{15}$

    so

    $f(x) = \dfrac{2}{15}(x^2 - x + 2),~x \in [-1,2]$

    integrating we get

    $F(x) = \displaystyle \int_{-\infty}^x~\dfrac{2}{15}(x^2 - x + 2)~dx =

    \begin{cases}
    0 &x<-1 \\
    \dfrac{1}{45} \left(2 x^3-3 x^2+12 x+17\right) &x\in [-1,2] \\
    1 &2 < x
    \end{cases}$
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  5. #5
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    Re: Integral calculus

    Thanks fur that romsek, I'm going to try and figure this all out not.

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  6. #6
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    Re: Integral calculus

    Sorry my post should have said - Thanks for that romsek, I'm going to try and figure this all out now.

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