Results 1 to 2 of 2

Math Help - Find the probability that Y<= 1?

  1. #1
    Yan
    Yan is offline
    Member
    Joined
    May 2008
    Posts
    103

    Find the probability that Y<= 1?

    Suppose Y is uniform on [0;X + 1], where X is Poisson with mean 3. Find the
    probability that Y <= 1.

    Can you show all your work? thanks
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Moo
    Moo is offline
    A Cute Angle Moo's Avatar
    Joined
    Mar 2008
    From
    P(I'm here)=1/3, P(I'm there)=t+1/3
    Posts
    5,618
    Thanks
    6
    Hello,
    Quote Originally Posted by Yan View Post
    Suppose Y is uniform on [0;X + 1], where X is Poisson with mean 3. Find the
    probability that Y <= 1.

    Can you show all your work? thanks
    \mathbb{P}(Y\leq 1)=\sum_{k=0}^\infty \mathbb{P}(Y\leq 1 \mid X=k)\mathbb{P}(X=k)=\sum_{k=0}^\infty \mathbb{P}(Y\leq 1 \mid X=k) \cdot\frac{3^k e^{-3}}{k!}

    But then, which distribution follows the random variable Y\mid X=k ? It's a uniform distribution over [0,k+1]
    Where the pdf is f(y)=\begin{cases}\frac{1}{k+1} & \text{ if } y\in[0,k+1] \\ 0 & \text{ otherwise} \end{cases}=\frac{1}{k+1} \cdot \bold{1}_{[0,k+1]}(y)

    \mathbb{P}(Y\leq 1\mid X=k)=\int_0^1 \frac{1}{k+1} ~dy (because k+1\geq 1 almost surely, we get these boundaries)

    Which is \mathbb{P}(Y\leq 1\mid X=k)=\frac{1}{k+1}


    Hence
    \begin{aligned}<br />
\mathbb{P}(Y\leq 1)<br />
&=\sum_{k=0}^\infty \frac{3^k e^{-3}}{(k+1)!} \\<br />
&=\frac 13 \sum_{k=0}^\infty \frac{3^{k+1}e^{-3}}{(k+1)!} \\<br />
&=\frac 13 \sum_{k=1}^\infty \frac{3^k e^{-3}}{k!} \\<br />
&=\frac 13 \left(\sum_{k=0}^\infty \frac{3^k e^{-3}}{k!}-e^{-3}\right)<br />
\end{aligned}

    But the sum is exactly the sum of probabilities of a Poisson distribution with mean 3. So it equals 1.

    Therefore,
    \mathbb{P}(Y\leq 1)=\frac 13 ~ (1-e^{-3})


    Looks good to you ?
    Last edited by Moo; May 29th 2009 at 11:17 AM. Reason: these not this
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Find a probability
    Posted in the Statistics Forum
    Replies: 4
    Last Post: December 9th 2009, 12:23 AM
  2. Find the indicated probability?
    Posted in the Statistics Forum
    Replies: 1
    Last Post: October 5th 2009, 09:35 PM
  3. Find the indicated probability?
    Posted in the Statistics Forum
    Replies: 7
    Last Post: June 8th 2009, 06:49 PM
  4. 2a. Find the Probability
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: February 7th 2008, 01:13 AM
  5. find the probability
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: April 16th 2007, 07:49 PM

Search Tags


/mathhelpforum @mathhelpforum