Joint Probability Question?

Question

I hate just asking the same questions from the assignment, but time's running out and I've never done anything like this before, so I know if I try to work through it and ask specific questions I'll get nowhere, but if I ask hopefully someone can explain. So:

The random variable X measures the concentration of ethanol in a chemical solution, and the random variable Y measures the acidity. They have a joint probability density function f(x,y) = A(20 – x – 2y) for 0<=x<=5 and 0<=y<=5 and f(x,y)=0 elsewhere.
A)What is the value of A?
B)What is P(1<=X<=2, 2<=Y<=3)?
C)Construct the marginal probability density functions fX(x) and fY(y)
D)Are the ethanol concentration and the acidity independent?
E)What are the expectation and the variance of the ethanal concentration?
F)What are the expectation and variance of the acidity?
G)If the ethanol concentration is 3, what is the conditional probability density function of the acidity?

Explanations, not just answers, please.

Answer 1

There's a lot of stuff here, and I've got to go to bed. At least I can get started on this without looking anything up.

A. The integral of the pdf has to equal 1, so

int f(x,y) dxdy over the range of x & y is
A[(20 - 2y)x - (x^2)/2] dy = A{[20x - (x^2)/2] - 2xy} dy
= A[20xy - (x^2)y/2 - xy^2]

and when you plug in the limits, you get

1 = A(20*25 - 125/2 - 125) = (A/2)(1000 - 125 - 250)

so A = 2/625 = 0.0032

B. You need the pdf integrated over the limits indicated. Your cumulative density function is

P(x,y < a,b) = .0032[20xy - (x^2)y/2 - xy^2]

Inserting the upper limits, you get

P(x,y < 2,3) = .0032(20*6 - 6 - 18) = 0.3072

The lower limits gives you

P(x,y < 1,2) = .0032(20*2 - 1 - 4) = 0.112

so the probability you want is the difference, P = 0.1952

C. To get the marginal pdf's, I'd take the cdf

P(x,y < a,b) = .0032[20xy - (x^2)y/2 - xy^2]

and, separately for x and y, add a "delta" (h) to get

P(x,y < a,b) = .0032{20(x+h)y - [(x+h)^2]y/2 - (x+h)y^2}

and then subtract the original function. You'll have some algebra here. (I showed it for x; here, hold y constant.)

The function P(x+h) - P(x) will give you the marginal probability for x.

D. There's no xy term in the original pdf, so I think x and y are independent. Would have to think about that some more, and probably look at a textbook.

E. I'd have to look this up.

F. For x = 3,

P(y) = A(20 - 3 - 2y) = A(17 - 2y)

Integrate from 0 to 5:

1 = A(17*5 - 25) ==> A = 1/60

P(y) = (17 - 2y)/60