Joint Probability Distribution [insert desperate hw help plead here]

[Shaq's Left Nut]

So I've worked on these problems for quite a while but I can't figure them out.
I'm not looking for someone to just do them for me, but if anyone can push me in the right direction or tell me what to go about integrating, etc. it would help

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

f(p,s) = 5pe^-ps, for 0.20 < p < 0.40, s > 0




1) The marginal distribution of p has the following distribution:
-discrete uniform
-continuous uniform
-binomial
-gamma
-beta

2) What is the mean price?

3) The conditional density of S given P has the following form:

4) What is the probability that the sales will be less than 30,000 units, given P = 25 cents?

5) What is the probability that the sales will be less than 30,000 units, given P is less than or equal to 25 cents?

6) What is the expected sales when P = 25 cents?


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Here is what I have so far, I don't know if any of these are right so don't assume they are, I'm just trying to show the effort is here.


1) f(p) = integral of f(p,s) over ds from 0 to infinity, comes out to f(p) = 5. I don't understand what that means, so I would guess the distribution is discrete uniform because we are dealing with cents.. (not sure its right)

2) To find the mean price, integrate f(p,s) over ds from 0 to infinity, which is f(p) = 5 again,
then E(p) = integrate p*f(p)dp over p from 0.20 to 0.40 = 30 cents (not sure its right)

3) f(s|p) = f(p,s)/f(p) = pe^-ps (sure its right)

4) P(s<3 | p=0.25) = pe^-ps = .25e^-.25s, integrate over S from 0 to 3 = 0.5276334473 (sure its right)

5) idk

6) idk



Any help is appreciated

[Shaq's Left Nut]

ok I can't figure out is #5, if anyone could push me in the right direction it would be appreciated