Final answer:
The student's problem involves finding a constant in a joint probability density function, calculating a specific probability, and determining densities, expected values, and whether two random variables are identically distributed or independent. Essential steps include integrating over appropriate bounds and comparing resulting functions.
Step-by-step explanation:
Finding the Constant c and Probability Tasks
To solve the given problem, we must first find the value of the constant c for the joint density function f(x, y) = c(x + y) in the domain 0 < x, y < 2.
(a) To find c, we must ensure that the integral of f(x, y) over the specified domain is equal to 1:
- Integrate f(x, y) over the domain for both x and y to get the total probability.
- Set this integral equal to 1 and solve for c.
(b) To calculate P(Y > X + 1), we:
- Set up the integral of f(x, y) over the region where y > x + 1 within the given domain.
- Calculate the integral to obtain the probability.
(c) The density of X, fX(x), is found by:
- Integrating f(x, y) with respect to y over the domain 0 < y < 2.
- This will yield the marginal probability density function of X.
(d) To determine if X and Y are identically distributed, we:
- Compare the marginal distributions fX(x) and fY(y).
- If they have the same distribution, they are identically distributed.
(e) To discuss if X and Y are independent, we:
- Check if the product of the marginal densities fX(x) * fY(y) equals f(x, y).
- If it does, X and Y are independent; otherwise, they are not.
(f) Finally, to find the expected value E(XY), we:
- Integrate the product x * y * f(x, y) over the domain for both x and y.
- Calculate this integral, which will yield the expected value of XY.
Note:
The integration bounds and constraints must be taken into account at each step to ensure appropriate limits of integration.