Results 1 to 2 of 2

Thread: distribution function with uniformity

  1. #1
    Senior Member
    Joined
    Jan 2008
    From
    Montreal
    Posts
    311
    Awards
    1

    distribution function with uniformity

    I'm having trouble solving the following problem correctly:

    The waiting time $\displaystyle Y$ until delivery of a new component for an industrial operation is uniformly distributed over the interval 1 to 5 days. The cost of this delay is given by $\displaystyle U = 2Y^2+3$. Find the probability density function for $\displaystyle U$.

    so far I have:

    $\displaystyle Y= \pm \sqrt{\frac{u-3}{2}}$

    and based on an example from the book I have:

    $\displaystyle f_U(u) = \left\{ \begin{array}{rcl}
    \frac{1}{2\sqrt{u}} \left[f_Y\left(\sqrt{u}\right) + f_Y(y)(-\sqrt{u})\right] & \mbox{for} & u>0 \\
    0 & \mbox{if} & \mbox{other}
    \end{array}\right.$

    then combining for the top part I would get:

    $\displaystyle =\frac{1}{2\sqrt{\frac{u-3}{2}}} \left[\frac{1}{4}\left(\sqrt{\frac{u-3}{2}}\right) + \frac{1}{4}\left(-\sqrt{\frac{u-3}{2}}\right)\right]$

    $\displaystyle =\frac{1}{8\sqrt{\frac{u-3}{2}}}$

    and based on the solution in the back of the book it's:

    $\displaystyle =\frac{1}{{\color{red}16}\sqrt{\frac{u-3}{2}}}$

    if anyone can help me it it would be greatly appreciated
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    9
    Quote Originally Posted by lllll View Post
    I'm having trouble solving the following problem correctly:

    The waiting time $\displaystyle Y$ until delivery of a new component for an industrial operation is uniformly distributed over the interval 1 to 5 days. The cost of this delay is given by $\displaystyle U = 2Y^2+3$. Find the probability density function for $\displaystyle U$.

    so far I have:

    $\displaystyle Y= \pm \sqrt{\frac{u-3}{2}}$

    and based on an example from the book I have:

    $\displaystyle f_U(u) = \left\{ \begin{array}{rcl}
    \frac{1}{2\sqrt{u}} \left[f_Y\left(\sqrt{u}\right) + f_Y(y)(-\sqrt{u})\right] & \mbox{for} & u>0 \\
    0 & \mbox{if} & \mbox{other}
    \end{array}\right.$

    then combining for the top part I would get:

    $\displaystyle =\frac{1}{2\sqrt{\frac{u-3}{2}}} \left[\frac{1}{4}\left(\sqrt{\frac{u-3}{2}}\right) + \frac{1}{4}\left(-\sqrt{\frac{u-3}{2}}\right)\right]$

    $\displaystyle =\frac{1}{8\sqrt{\frac{u-3}{2}}}$

    and based on the solution in the back of the book it's:

    $\displaystyle =\frac{1}{{\color{red}16}\sqrt{\frac{u-3}{2}}}$

    if anyone can help me it it would be greatly appreciated
    $\displaystyle F(u) = \Pr(U \leq u)$

    $\displaystyle = \Pr(2Y^2 + 3 \leq u) = \Pr \left( Y^2 \leq \frac{u-3}{2} \right)$ $\displaystyle = \Pr \left( - \sqrt{\frac{u-3}{2}} \leq Y \leq \sqrt{\frac{u-3}{2}} \right)$


    $\displaystyle = \int_{1}^{\sqrt{\frac{u-3}{2}}} \frac{1}{4} \, dy$


    $\displaystyle = \frac{1}{4} \sqrt{\frac{u-3}{2}} - \frac{1}{4}$.


    Therefore $\displaystyle f(u) = \frac{dF}{du} = \frac{1}{16\sqrt{\frac{u-3}{2}}}$ for $\displaystyle 5 \leq u \leq 53$ and zero otherwise.


    Note: You can see your answer is wrong because if you integrate it between u = 5 and u = 53 you get 2 rather than 1.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Finding a Cumulative Distribution Function and Density Function
    Posted in the Advanced Statistics Forum
    Replies: 3
    Last Post: Nov 17th 2011, 09:41 AM
  2. cumulative distribution function using a gamma distribution
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: Sep 11th 2010, 10:05 AM
  3. Distribution function for a function of random variable
    Posted in the Advanced Statistics Forum
    Replies: 3
    Last Post: Apr 8th 2010, 12:14 PM
  4. Moment generating function and distribution function?
    Posted in the Advanced Statistics Forum
    Replies: 4
    Last Post: Jan 25th 2009, 02:02 PM
  5. Cumulative distribution function of binomial distribution
    Posted in the Advanced Statistics Forum
    Replies: 2
    Last Post: Oct 31st 2008, 03:34 PM

Search tags for this page

Click on a term to search for related topics.

Search Tags


/mathhelpforum @mathhelpforum