# Joint Probability density

• Jan 24th 2009, 10:30 AM
Yan
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.
• Jan 24th 2009, 01:07 PM
mr fantastic
Quote:

Originally Posted by Yan
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) $\displaystyle \int_{s=0}^{s=3} f_S(s | p = 0.25) \, ds$.
• Jan 25th 2009, 10:17 AM
Yan
Quote:

Originally Posted by mr fantastic
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?
• Jan 25th 2009, 12:48 PM
mr fantastic
Quote:

Originally Posted by Yan
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 "?!

..
• Jan 25th 2009, 05:22 PM
Yan
how to calculate the first part (the ds part, http://www.mathhelpforum.com/math-he...1f17a78b-1.gif. 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 http://www.mathhelpforum.com/math-he...cf451136-1.gif, 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
• Jan 25th 2009, 06:10 PM
mr fantastic
Quote:

Originally Posted by Yan
[snip]
and I think the dp part is http://www.mathhelpforum.com/math-he...cf451136-1.gif, 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
how to calculate the first part (the ds part, http://www.mathhelpforum.com/math-he...1f17a78b-1.gif. 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]

http://www.mathhelpforum.com/math-he...1f17a78b-1.gif 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.