a little bit rusty on statistics, i generally know how to set up the problems but not how to solve. thanks for any help you can give, especially with the integration bounds I'm confused with.
X & Y have a joint distribution, and the following is the joint density fnct:
f(x,y)={ 24xy for x>=0, y>=0, x+y<=1
{0 otherwise
*Compute marginal distribution and cumulative distribution of X:
f(x) = intg[over y] f(x,y)dy = 12xy^2?
F(x) = intg[over x] f(x)dx = 6x^2y^2?
*Determine whether X and Y are independent: f(x,y)=f(x)f(y) identity for independence
To compute the marginal distribution and cumulative distribution of X, you need to integrate the joint density function over the appropriate variable(s).
1. Marginal distribution of X (f(x)):
To find the marginal distribution of X, you need to integrate the joint density function f(x,y) with respect to y and evaluate it at x.
f(x) = ∫[0 to 1-x] 24xy dy
You integrate the joint density function over the feasible range of y, which is from 0 to 1-x because of the constraint x+y<=1. By integrating with respect to y, you are "marginalizing out" the y variable and getting the distribution of X alone.
Evaluating the integral:
f(x) = 24x * ∫[0 to 1-x] y dy
= 24x * (y^2/2) |[0 to 1-x]
= 12x(1-x)^2
Therefore, the marginal distribution of X is f(x) = 12x(1-x)^2.
2. Cumulative distribution of X (F(x)):
To find the cumulative distribution of X, you integrate the marginal distribution function f(x) with respect to x.
F(x) = ∫[0 to x] f(t) dt
Using the marginal distribution function we found previously:
F(x) = ∫[0 to x] 12t(1-t)^2 dt
To evaluate this integral, you would need to integrate it by parts or use tabulated integrals.
Regarding independence:
To determine whether X and Y are independent, you need to check if the joint density function can be expressed as the product of the marginal density functions.
If X and Y are independent, then f(x,y) = f(x) * f(y) for all x and y.
In this case, you can compare the joint density function f(x,y) = 24xy with the product of the marginal density functions f(x) = 12x(1-x)^2 and f(y) = 24y.
By comparing f(x,y) with f(x)f(y), you can see that f(x,y) is not equal to f(x)f(y), so X and Y are not independent.
I hope this explanation helps you understand how to solve the problem and deal with integration bounds in statistics. Let me know if you have any further questions!