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Math Help - Probability Density Functions

  1. #1
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    Probability Density Functions

    Hi, sorry need help on a second problem. I've figured out the probability here as

    0.68 by integrating from 1 to 1.2 for 2-x and add to integration from 0.8 to 1 for x but I'm not sure if I'm using the distribution function or th probability density

    Question

    f(x) x for 0<x<1
    2-x for 1<=x<2
    0 elsewhere

    Find P(0.8<X<1.2) using
    a. The probability density
    b. the distribution function

    The way I did it I think seems to have used the distribution function. How would I find the same result with the probability density?
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  2. #2
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    F(x)=\int_{0}^{x}xdx, \;\ 0 \;\ to \;\ 1

    F(x)=\int_{0}^{1}xdx=\frac{1}{2}

    F(x)=\int_{1}^{2}(2-x)dx

    \frac{1}{2}+\int_{1}^{x}(2-x)dx=2x-\frac{x^{2}}{2}-1

    F(x)=\begin{Bmatrix}0, \;\ x\leq 0\\ \frac{x^{2}}{2}, \;\ 0<x<1\\ 2x-\frac{x^{2}}{2}-1, \;\ 1\leq x<2\\ 1, \;\ 2\leq x\end{Bmatrix}

    P(.8 < x < 1.2)=\int_{.8}^{1}xdx+\int_{1}^{1.2}(2-x)dx
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  3. #3
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    Quote Originally Posted by galactus View Post
    F(x)=\int_{0}^{x}xdx, \;\ 0 \;\ to \;\ 1

    F(x)=\int_{0}^{1}xdx=\frac{1}{2}

    F(x)=\int_{1}^{2}(2-x)dx

    \frac{1}{2}+\int_{1}^{x}(2-x)dx=2x-\frac{x^{2}}{2}-1

    F(x)=\begin{Bmatrix}0, \;\ x\leq 0\\ \frac{x^{2}}{2}, \;\ 0<x<1\\ 2x-\frac{x^{2}}{2}-1, \;\ 1\leq x<2\\ 1, \;\ 2\leq x\end{Bmatrix}

    P(.8 < x < 1.2)=\int_{.8}^{1}xdx+\int_{1}^{1.2}(2-x)dx
    Adding one thing:

    Using what galactus posted to calculate \Pr(0.8 < X < 1.2) = F(1.2) - F(0.8) is the distribution approach requested in part (b) of the question.
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  4. #4
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    Quote Originally Posted by mr fantastic View Post
    Adding one thing:

    Using what galactus posted to calculate \Pr(0.8 < X < 1.2) = F(1.2) - F(0.8) is the distribution approach requested in part (b) of the question.
    Right, that's what I thought too. The book also wants us to use the density function approach, is there a way to do that?
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  5. #5
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    haha, this is too funny. Apparently for both ways you do it exactly the same, just with distribution function you make a chart of the distribution first
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  6. #6
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    Yes, the distribution is the chart I showed. The probability density is the integration.
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