Results 1 to 9 of 9

Math Help - Poisson probability

  1. #1
    Eater of Worlds
    galactus's Avatar
    Joined
    Jul 2006
    From
    Chaneysville, PA
    Posts
    3,001
    Thanks
    1

    Poisson probability

    Hi All:

    This time I am asking a question. I am teaching stats next semester after several years of not doing it. I was looking through an old stats book and doing some problems. Here is a Poisson that has me stumped.

    "Cars going through an intersection average 40 per hour. What is the probability that the time between the arrival of a car and the second car after it is at least 2 minutes?"

    In this case, they are asking for 'at least 2 MINUTES", but the average is given in hours. So, {\lambda}=\frac{40}{60}=\frac{2}{3}

    I was thinking maybe a gamma, like so:

    \frac{1}{(\frac{2}{3})^{2}}\int_{2}^{\infty}te^{\f  rac{-t}{\frac{2}{3}}}dt=.199

    but that does not seem correct.



    \frac{2}{3}\int_{2}^{\infty}e^{\frac{-2}{3}t}dt=e^{\frac{-4}{3}}

    But this would be fine if the problem said, "what is the probability that the time between arrivals of any two cars is at least two minutes.

    It's the "arrival of a car and the second after that is at least two minutes" I am unsure of.

    Any ideas what I am overlooking. I feel rather obtuse on this. MrF, what is my oversight/mental block?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member Focus's Avatar
    Joined
    Aug 2009
    Posts
    228
    Quote Originally Posted by galactus View Post
    I was thinking maybe a gamma, like so:

    \frac{1}{(\frac{2}{3})^{2}}\int_{2}^{\infty}te^{\f  rac{-t}{\frac{2}{3}}}dt=.199

    but that does not seem correct.



    \frac{2}{3}\int_{2}^{\infty}e^{\frac{-2}{3}t}dt=e^{\frac{-4}{3}}

    But this would be fine if the problem said, "what is the probability that the time between arrivals of any two cars is at least two minutes.

    It's the "arrival of a car and the second after that is at least two minutes" I am unsure of.

    Any ideas what I am overlooking. I feel rather obtuse on this. MrF, what is my oversight/mental block?
    Well assuming that the cars arrive independently, this forms a Poisson process with the rate given. Poisson process has a holding time that is exponential with the same rate, and the holdings are independent both of each other and the process.

    Hope this helps.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Eater of Worlds
    galactus's Avatar
    Joined
    Jul 2006
    From
    Chaneysville, PA
    Posts
    3,001
    Thanks
    1
    I think I got it.

    \text{P(0 or 1 car passes by in 2 minutes)}

    e^{-{\lambda}t}+e^{-{\lambda}t}\cdot {\lambda}t={\lambda}^{2}\int_{2}^{\infty}te^{-{\lambda}t}dt

    When {\lambda}=\frac{2}{3}, then

    \frac{7}{3}e^{-\frac{4}{3}}\approx .615
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member Focus's Avatar
    Joined
    Aug 2009
    Posts
    228
    By a holding time I mean how long it stays constant. Say that \mathbf{e}_\lambda is an exponential r.v. with parameter \lambda. So the probability you are looking for is \mathbb{P}(\mathbf{e}_\lambda \geq 2)=e^{-2\lambda}=e^{-4/3}\approx 0.263.

    Your last answer is obviously incorrect as the mean, which is 1.5 minutes between cars, is greater than the median. 2 minutes or more should have a probability less than a half.

    I was trying to support your initial line of thinking. As the cars arrive independently, we wait an exponential time between any of them. The answer is the same for any two cars.

    Disclaimer: I may be wrong as I am starting to confuse myself as well
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Eater of Worlds
    galactus's Avatar
    Joined
    Jul 2006
    From
    Chaneysville, PA
    Posts
    3,001
    Thanks
    1
    Thank you for your response.

    The probability I got for "what is the probability that the arrival time between any two cars is at least two minutes?" is what you have \frac{2}{3}\int_{2}^{\infty}e^{\frac{-2}{3}t}dt=.263

    But, for "what is the probability that the arrival time between a car and the second car after it is at least two minutes?", I got:

    1-P(\text{0 or 1 car arrives in 2 minutes})

    1-\frac{2}{3}\cdot \frac{2}{3}\int_{2}^{\infty}te^{\frac{-4t}{3}}dt=.385

    Perhaps I am wrong, but it seems logical.
    Last edited by galactus; December 15th 2009 at 08:23 AM.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Super Member
    Joined
    Mar 2008
    Posts
    934
    Thanks
    33
    Awards
    1
    Quote Originally Posted by galactus View Post
    Thank you for your response.

    The probability I got for "what is the probability that the arrival time between any two cars is at least two minutes?" is what you have \frac{2}{3}\int_{2}^{\infty}e^{\frac{-2}{3}t}dt=.263

    But, for "what is the probability that the arrival time between a car and the second car after it is at least two minutes?", I got:

    1-P(\text{0 or 1 car arrives in 2 minutes})

    1-\frac{2}{3}\cdot \frac{2}{3}\int_{2}^{\infty}te^{\frac{-4t}{3}}dt=.385

    Perhaps I am wrong, but it seems logical.
    The time between cars has an Exponential distribution with \lambda = 2/3 per minute. You want

    P(X > 2) where X has that distribution.

    For an Exponential distribution,
    P(X > x) = e ^ {-\lambda x}

    So...
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Eater of Worlds
    galactus's Avatar
    Joined
    Jul 2006
    From
    Chaneysville, PA
    Posts
    3,001
    Thanks
    1
    Yeah, that makes more sense.

    \int_{2}^{\infty}e^{\frac{-4}{3}}dt
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Super Member
    Joined
    Mar 2008
    Posts
    934
    Thanks
    33
    Awards
    1
    Quote Originally Posted by galactus View Post
    Yeah, that makes more sense.

    \int_{2}^{\infty}e^{\frac{-4}{3}}dt
    No integral, just e^{-4/3}.
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Eater of Worlds
    galactus's Avatar
    Joined
    Jul 2006
    From
    Chaneysville, PA
    Posts
    3,001
    Thanks
    1
    Oh, the 'at least 2' threw me. I have to look at these more. I used to know these, but it's been a while. Thanks. It's coming back, but I am embarrassed. I had to prove the derivation of this some years back, believe it or not.

    What messed me up was this is the same answer I got for 'the probability that the arrival time of any two cars is at least two minutes".
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Probability - Poisson
    Posted in the Statistics Forum
    Replies: 3
    Last Post: November 8th 2010, 02:19 PM
  2. Poisson Probability
    Posted in the Advanced Statistics Forum
    Replies: 3
    Last Post: June 21st 2010, 08:37 AM
  3. Poisson Probability
    Posted in the Statistics Forum
    Replies: 2
    Last Post: February 20th 2010, 09:14 AM
  4. poisson probability
    Posted in the Advanced Statistics Forum
    Replies: 11
    Last Post: October 10th 2008, 03:53 PM
  5. Poisson Probability
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: August 2nd 2008, 09:34 AM

Search Tags


/mathhelpforum @mathhelpforum