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Math Help - Difficult Question Help (Probability/Statistics) Joint Probability Distributions

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    Difficult Question Help (Probability/Statistics) Joint Probability Distributions

    Hi Guys,

    Hope everyone is well. I'm wondering if someone can help me with a difficult problem presented during lecture today.

    Two friends plan to meet to go to a nightclub. Each of them arrives at a time uniformly distributed between midnight and 1am and independently of the other. Denote by X (respectively Y) the random variable representing the arrival time of the first person (respectively, the second). The joint probability distribution is given by

    f_(x,y) (x,y) = 2 if 0 ≤ x ≤ y, and 0 otherwise.

    a) Find the probability that the first person is waiting for his friend for more than 10 minutes.

    b) Determine the marginal probability density functions of X and Y. Check that they are indeed probability density functions.

    c) Calculate the means E[X] and E[Y].

    d) Calculate the variances V(X) and V(Y).

    e) Find the conditional density function of X given that Y = y, for 0 ≤ y ≤ 1. Check that it is indeed a probability density function.

    f) Repeat part (e) for the conditional density of Y given that X = x.
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  2. #2
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    Re: Difficult Question Help (Probability/Statistics) Joint Probability Distributions

    Quote Originally Posted by Radiance View Post
    Hi Guys,

    Hope everyone is well. I'm wondering if someone can help me with a difficult problem presented during lecture today.

    Two friends plan to meet to go to a nightclub. Each of them arrives at a time uniformly distributed between midnight and 1am and independently of the other. Denote by X (respectively Y) the random variable representing the arrival time of the first person (respectively, the second). The joint probability distribution is given by

    f_(x,y) (x,y) = 2 if 0 ≤ x ≤ y, and 0 otherwise.

    a) Find the probability that the first person is waiting for his friend for more than 10 minutes.

    b) Determine the marginal probability density functions of X and Y. Check that they are indeed probability density functions.

    c) Calculate the means E[X] and E[Y].

    d) Calculate the variances V(X) and V(Y).

    e) Find the conditional density function of X given that Y = y, for 0 ≤ y ≤ 1. Check that it is indeed a probability density function.

    f) Repeat part (e) for the conditional density of Y given that X = x.
    Assuming y \leq 1 (initial problem did not state this, but it's safe to assume).

    a) Since 10 min means X=1/6, we want to find P(X>1/6)=1-P(x<1/6) = 1-\int_0^{1/6} \int_0^x f(x,y) dy dx
    b) f_X(X) = \int_0^x f(x,y) dy and f_Y(Y) = \int_0^y f(x,y) dx, check that for both f's that the probability functions (the integrals of the f's) are equal to 1.

    Once you have a) and b), calculating c)-f) should be straightforward enough ... you'll use the results of a) and b) (just use the definitions of expectancy, variance, conditional prob).
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