Let U, V be iid Unif(0, 1) random variables, and set
M = max(U,V) and N = min (U,V)
(a) Find the conditional density of N given M = a for any value
of a ∈ (0, 1).
(b) Find Cov(M, N).
Let z be a random variable with a standard normal distribution.
Find P(0 ≤ z ≤ 0.46), and shade the corresponding area under the
standard normal curve. (Use 4 decimal places.)
Let Zt = U sin(2*pi*t) + V cos(2*pi*t), where U and V are
independent random variables, each with
mean 0 and variance 1.
(a) Is Zt strictly stationary?
(b) Is Zt weakly stationary?
Let X; be n IID U(0, 1) random variables. What are the mean and
variance of the minimum-order and maximum-order statistics?
PLEASE SHOW ALL WORK AND FORMULAS USED
1. Let X and Y be independent U[0, 1] random variables, so that
the point (X, Y) is uniformly distributed in the unit square.
Let T = X + Y.
(a) Find P( 2Y < X ).
(b). Find the CDF F(t) of T (for all real numbers t).
HINT: For any number t, F(t) = P ( X <= t) is just the area
of a part of the unit square.
(c). Find the density f(t).
REMARK: For a...
Let X and Y be independent and uniformly distributed random
variables on [0, 1]. Find the cumulative distribution and
probability density function of Z = X + Y.
Let Y and Z be independent continuous random variables, both
uniformly distributed between 0 and 1.
1. Find the CDF of |Y − Z|.
2. Find the PDF of |Y − Z|.