Results 1 to 3 of 3

Math Help - Joint Probability

  1. #1
    Junior Member
    Joined
    Dec 2007
    Posts
    32

    Joint Probability

    Hi
    I've been looking at this problem for awhile now.

    Annie and Alvie have agreed to meet between 5:00 pm and 6:00 pm for dinner at a local health food restaurant. Let X=Annie's arrival time and Y=Alvie's arrival time. Suppose X and Y are independent with each uniformly distributed on the interval [5, 6].

    a. What is the joint pdf of X and Y?
    b. What is the probability that they both arrive between 5:15 and 5:45?
    c. If the first one to arrive will wait only 10 minutes before leaving to eat elsewhere, what is the probability that they have dinner at the health-food restaurant? [Hint: The event of interest is A={(x,y): | x-y | ≤ 1/6}.

    For [A]

    I set the pdf of x & y = (1/60) for 5 hr, 0 min <= x,y <= 5 hr, 60 min / 0 otherwise

    for the pdf of f(x,y) = (1/3600) for 5 hr, 0 min <= x,y <= 5 hr, 60 min / 0 otherwise

    Is this correct? If not, please explain.

    Can you also show me how I can set up parts [B] & [C]
    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
    5
    Quote Originally Posted by shogunhd View Post
    Hi
    I've been looking at this problem for awhile now.

    Annie and Alvie have agreed to meet between 5:00 pm and 6:00 pm for dinner at a local health food restaurant. Let X=Annie's arrival time and Y=Alvie's arrival time. Suppose X and Y are independent with each uniformly distributed on the interval [5, 6].

    a. What is the joint pdf of X and Y?
    b. What is the probability that they both arrive between 5:15 and 5:45?
    c. If the first one to arrive will wait only 10 minutes before leaving to eat elsewhere, what is the probability that they have dinner at the health-food restaurant? [Hint: The event of interest is A={(x,y): | x-y | ≤ 1/6}.

    For [A]

    I set the pdf of x & y = (1/60) for 5 hr, 0 min <= x,y <= 5 hr, 60 min / 0 otherwise

    for the pdf of f(x,y) = (1/3600) for 5 hr, 0 min <= x,y <= 5 hr, 60 min / 0 otherwise

    Is this correct? If not, please explain.

    Can you also show me how I can set up parts [b] & [C]
    Since you're told that X and Y are:

    1. independent, and
    2. each is uniformly distributed on the interval [5, 6]

    the joint pdf is f(x, y) = g(x) g(y) = \left( \frac{1}{6 - 5} \right) \left( \frac{1}{6 - 5} \right) = 1 for 5 \leq x \leq 6, 5 \leq y \leq 6 and zero otherwise.

    I'm out of time now - will come back later to look at (b) and (c) unless someone else replies in the meantime.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by shogunhd View Post
    Hi
    I've been looking at this problem for awhile now.

    Annie and Alvie have agreed to meet between 5:00 pm and 6:00 pm for dinner at a local health food restaurant. Let X=Annie's arrival time and Y=Alvie's arrival time. Suppose X and Y are independent with each uniformly distributed on the interval [5, 6].

    a. What is the joint pdf of X and Y?
    b. What is the probability that they both arrive between 5:15 and 5:45?
    c. If the first one to arrive will wait only 10 minutes before leaving to eat elsewhere, what is the probability that they have dinner at the health-food restaurant? [Hint: The event of interest is A={(x,y): | x-y | ≤ 1/6}.

    For [A]

    I set the pdf of x & y = (1/60) for 5 hr, 0 min <= x,y <= 5 hr, 60 min / 0 otherwise

    for the pdf of f(x,y) = (1/3600) for 5 hr, 0 min <= x,y <= 5 hr, 60 min / 0 otherwise

    Is this correct? If not, please explain.

    Can you also show me how I can set up parts [b] & [C]
    Quote Originally Posted by mr fantastic View Post
    Since you're told that X and Y are:

    1. independent, and
    2. each is uniformly distributed on the interval [5, 6]

    the joint pdf is f(x, y) = g(x) g(y) = \left( \frac{1}{6 - 5} \right) \left( \frac{1}{6 - 5} \right) = 1 for 5 \leq x \leq 6, 5 \leq y \leq 6 and zero otherwise.
    [snip]
    b. \Pr(1/4 \leq X \leq 3/4, 1/4 \leq Y \leq 3/4) = \int_{y = 1/4}^{y = 3/4} \int_{x = 1/4}^{x = 3/4} 1 \, dx \, dy = ....


    c. You need to integrate f(x, y) over the region R_{xy} which I'll describe below (I'm too lazy to draw a diagram etc.):

    Consider the square in the xy-plane bounded by the coordinate axes and the lines y = 1, x = 1. Note that |x - y| = 1/6 => y = x - 1/6 or y = x + 1/6. The region R_{xy} is the area of the square that is between these two lines ....

    If you have trouble setting up the necessary double integral, please say so. To do it you'll need to divide the region into three subregions .....
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Joint probability
    Posted in the Advanced Statistics Forum
    Replies: 2
    Last Post: May 31st 2010, 10:57 AM
  2. Joint Probability/Marginal Probability
    Posted in the Statistics Forum
    Replies: 0
    Last Post: March 2nd 2010, 10:15 PM
  3. Replies: 5
    Last Post: December 5th 2009, 11:30 PM
  4. Joint Probability
    Posted in the Advanced Statistics Forum
    Replies: 2
    Last Post: March 12th 2009, 11:53 PM
  5. Joint Probability
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: April 27th 2008, 08:09 PM

Search Tags


/mathhelpforum @mathhelpforum