solve please

7. Suppose that the joint probability density function of two random variables X and Y is Ax, y) = c(xty) for x between 0 and 2, and for y between 0 and 1, and zero off of this rectangle. Determine the conditional density of Y given X. (Note that you will have to work out the value of the constant c to complete this problem.) 8. Suppose that the random variables X1, Xz, ... X, form a random sample of size n, with the marginal density of each X, equal to f(x) = ex for x greater than or equal to 1 (and zero otherwise). Find the expected value of the minimum of the sample. 9. Suppose that X, Y, and Z are three random variables with expectations, variances, respectively 1, 4, and 8, and suppose that the covariance between X and Y is -1, the covariance between X and Z is 1, and the covariance between Y and Z is 2. Determine the variance of X + Y+ Z. 10. Suppose that the proportion 0 of defective items in a large manufactured lot is unknown, and the prior density of 0 is f(x) = 2x. When eight items are selected at random from the lot, it is found that exactly three of them are defective. Determine the posterior distribution of 0