Problem 1. Let x and y be two independent uniform random variables over the interval [0, 2]. Let z be the random variable defined by

xy

z =

x + y

The random variable z is the resistor corresponding to placing two resistors in parallel which are independent and uniformly  distributed over  the  interval  [0, 2].  You can  use  Matlab  or  any  other  program to compute the integrals in this problem. In fact, computing these integrals by hand will be very tedious and is not necessary.

Let H₂ and H₄ be the Hilbert spaces defined by

H₂ = span {1, z} and H₄ = span {1, z, z , z }

Let

x₂ = P_H2 x = α + βz

2 3

x^₄ = P_H4 x = a + bz + cz  + dz

1. Find and plot the density function f_z(z) for 0 ≤ z ≤ 1.

2. Find  g(z) = E(x z).  The formula for g(z)  = E(x z) is long and messy. If you cannot find g(z) try to find g(z) = E(x z = z)

numerically. You will need to plot g(z). If your plot of g(z) is correct we will give full credit for g^(z).

3. Find x^₂ = P_H2 x = α + βz

4. Find x^₄ = P_H4 x = a + bz + cz

+ dz³

2 3

5. Plot g(z) and (x₂)(z) = α + βz and (x₄)(z) = a + bz + cz

all on the same graph over the interval [0, 1].

+ dz