Suppose that X, Y , and Z are random variables with a joint density f(x, y, z) = ( 6/((1+x+y+z)^4)) , when x, y, z > 0, and 0, otherwise. Determine the distribution of X + Y + Z.

Suppose an agent has $100. He opens a demand deposit of $100 with a bank which has asset x where x is a random variable. a)         Suppose x is uniformly distributed on the interval [100, 200]. The density of x is f(x)=1/100 on [100,200] and f(x)=0 otherwise. What is the expected loss of the depositor?  b)         Suppose changes in the economy changes the distribution of x. Now x is uniformly distributed on the interval [40, 200]. The density of x is f(x)=1/160 on [40, 200] and f(x)=0 otherwise. What is the expected loss of the depositor?  c)         Suppose x is uniformly distributed on the interval [0, 200]. The density of x is f(x)=1/200 on [0, 200] and f(x)=0 otherwise. Calculate the expected loss of the depositor.  d)        Suppose x is uniformly distributed on the interval [0, 200]. Given the macroeconomic environment, the government introduces deposit insurance. There is deposit insurance of an amount I=80. Calculate the expected loss of the depositor.

If X is a continuous random variable with X ∼ Uniform([0, 2]), what is E[X^3]?

Suppose W Z = b+aU. N(u1, o}), and U ~ N(42, 03) with Cov(W,U) = 012. Let Y = a + bW and (1) Describe the distribution of Y . (2) Describe the distribution of Y + Z and Y – Z. (3) Find C'ov(Y, Z) and Cov(Y +Z,Y – Z).