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Thread: Joint Probability density

  1. #1
    Yan
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    Joint Probability density

    Suppose that P, the price of a certain commodity (in dollars), and S, its total sales (in 10,000 units), are random variables whose joint probability distribution can be approximated closely with the joint probability density

    f(p,s)=5pe^(-ps) for 0.2<p<0.4, s>0 and 0 elsewhere

    Find the probabilities that
    (a) the price will be less than 30 cents and sales will exceed 20,000 units;
    (b) the price will between 25 cents and 30 cents and sales will be less than 10,000 units;
    (c) the marginal density of P;
    (d) the conditional density of S given P=p;
    (e) the probability that sales will be less than 30,000 units when p=25 cents.
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    Quote Originally Posted by Yan View Post
    Suppose that P, the price of a certain commodity (in dollars), and S, its total sales (in 10,000 units), are random variables whose joint probability distribution can be approximated closely with the joint probability density

    f(p,s)=5pe^(-ps) for 0.2<p<0.4, s>0 and 0 elsewhere

    Find the probabilities that
    (a) the price will be less than 30 cents and sales will exceed 20,000 units;
    (b) the price will between 25 cents and 30 cents and sales will be less than 10,000 units;
    (c) the marginal density of P;
    (d) the conditional density of S given P=p;
    (e) the probability that sales will be less than 30,000 units when p=25 cents.
    These are all set up and solved form the basic definitions.

    (a) $\displaystyle \int_{p=0}^{p = 0.3} \int_{s=2}^{s=+\infty} f(p, s) \, ds \, dp$.


    (b) $\displaystyle \int_{p=0.25}^{p = 0.3} \int_{s=0}^{s=1} f(p, s) \, ds \, dp$.


    (c) $\displaystyle f_P(p) = \int_{s=0}^{s=+\infty} f(p, s) \, ds$.


    (d) $\displaystyle f_S(s | p) = \frac{f(p, s)}{f_P(p)}$.


    (e) $\displaystyle \int_{s=0}^{s=3} f_S(s | p = 0.25) \, ds$.
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    Yan
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    Quote Originally Posted by mr fantastic View Post
    These are all set up and solved form the basic definitions.

    (a) $\displaystyle \int_{p=0}^{p = 0.3} \int_{s=2}^{s=+\infty} f(p, s) \, ds \, dp$.
    how to calculate the first part (the ds part, $\displaystyle \int_{s=2}^{s=+\infty} f(p, s) \, ds$.
    and I think the dp part is $\displaystyle \int_{p=0.2}^{p=0.3} f(p, s) \, dp$, is it right?
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    Quote Originally Posted by Yan View Post
    how to calculate the first part (the ds part, $\displaystyle \int_{s=2}^{s=+\infty} f(p, s) \, ds$. Mr F says: Do you know how to integrate? You're integrating a simple exponential function. Where are you stuck here?

    and I think the dp part is $\displaystyle \int_{p=0.2}^{p=0.3} f(p, s) \, dp$, is it right? Mr F says: Why would you think that when the question clearly says "the price will be less than 30 cents "?!
    ..
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    Yan
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    how to calculate the first part (the ds part, . Mr F says: Do you know how to integrate? You're integrating a simple exponential function. Where are you stuck here? the problem is the S is form 2 to infin. there is not exactly number for infin. like if it is from negative infin to 2,then i know the number is from 0 to 2.

    and I think the dp part is , is it right? Mr F says: Why would you think that when the question clearly says "the price will be less than 30 cents "?! because the problem is the Price is 0.2<p<0.4, so think is should be 0.2 to 0.3
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    Quote Originally Posted by Yan View Post
    [snip]
    and I think the dp part is , is it right? Mr F says: Why would you think that when the question clearly says "the price will be less than 30 cents "?! because the problem is the Price is 0.2<p<0.4, so think is should be 0.2 to 0.3
    Well, that's a good reason why.

    Quote Originally Posted by Yan View Post
    how to calculate the first part (the ds part, . Mr F says: Do you know how to integrate? You're integrating a simple exponential function. Where are you stuck here? the problem is the S is form 2 to infin. there is not exactly number for infin. like if it is from negative infin to 2,then i know the number is from 0 to 2.

    [snip]
    is an improper integral. To find it you need to do the usual thing and consider a limit: $\displaystyle \lim_{a \rightarrow +\infty} \int_2^{a} f(p, s) \, ds$ etc.
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