3)
If we have some X, Y, Z such that I(X;Z) = 0 (which means X and Z are independent variables), then
I(X; YZ) = I(X; YZ)
I(X; Y) + I(X; Z|Y ) = I(X;Z) + I(X; Y |Z)
I(X; Y) + I(X; Z|Y) = I(X; Y|Z)
so I(X; Y|Z) − I(X;Y ) = I(X;Z|Y ) ≥ 0, which implies
I(X; Y |Z) ≥ I(X; Y).
Hence, that if any two of the three random variables X, Y, and Z are independent, I(X; Y) ≤ I(X; Y|Z) holds.
3, (20%) Prove that if any two of the three random variables X, Y, and Z are independent, I(X; Y)...
Suppose X, Y and Z are three different random variables. Let X obey Bernoulli Distribution. The probability distribution function is p(x) = Let Y obeys the standard Normal (Gaussian) distribution, which can be written as Y ∼ N(0, 1). X and Y are independent. Meanwhile, let Z = XY . (a) What is the Expectation (mean value) of X? (b) Are Y and Z independent? (Just clarify, do not need to prove) (c) Show that Z is also a standard...
I need help on 6.26 and 6.28 please! 6.26 Three independent continuous random variables X, Y, and Z are -uniformly distributed between 0 and 1 . Ifthe random variable S X+ Y+Z, determine the PDF of S. Suppose X and Y are two continuous random variables with the joint PDF fxr(x,y). Let the functions U and Wbe defined as follows: U w=X +2Y. Find the joint PDF fuwlu,w) 6.27 2X+3Y, and 6.28 Find fuw(u, w) in terms of fxrtx,y) if...
Problem 3. Let X and Y be two independent random variables taking nonnegative integer values (a) Prove that for any nonnegative integer m 7m k=0 b) Suppose that X~ B (n, p) and Y ~ B(m. p), and X, Y are independent. What is the distribution of the random variable Z X + Y? (c) Prove the following formula for binomial coefficients: n\ _n + m for kmin (m, n) (d) Let X ~ B (n, 1/2). What is P...
Prove that any two independent random variables are uncorrelated.
1 Expectation, Co-variance and Independence [25pts] Suppose X, Y and Z are three different random variables. Let X obeys Bernouli Distribution. The probability disbribution function is 0.5 x=1 0.5 x=-1 Let Y obeys the standard Normal (Gaussian) distribution, which can be written as Y are independent. Meanwhile, let Z = XY. N(0,1). X and Y (a) What is the Expectation (mean value) of X? 3pts (b) Are Y and Z independent? (Just clarify, do not need to prove) [2pts c)...
Let X and Y be two discrete random independent random variables. p(x) = 1/3 for x =-2,-1,0 p(y) = 1/2 for y =1,6 Z = X + Y. What is the distribution of Z using the method of MGF's
Question 1 、 Let X, Y and Z be three random variables that take values in the alphabet {0,1, M-lj. We assume X and Z are independent and Y = X +2(mod M), The distribution of Z is given as P(Z 0)1 -p and P (Z =i)= , for i = 1, M-1. For question 1-3 we M-1 will assume that X is uniform on f0,1,..,M-1}. Find H(X) and H(Z) Find H(Y ) Find 1 (X; Y) and「X, YZ) and...
x, y, and z are three independent Poisson random variables with the same mean. If P(x=1)=2P(x=0), what is P(x+y+z<=2)
Suppose three random variables X, Y, Z have a joint distribution PX,Y,Z(x,y,z)=PX(x)PZ∣X(z∣x)PY∣Z(y∣z). Then X and Y are independent given Z? True or False Suppose random variables X and Y are independent given Z , then the joint distribution must be of the form PX,Y,Z(x,y,z)=h(x,z)g(y,z), where h,g are some functions? True or false
You are given three independent random variables X, Y, and Z, all distributed exponentially, such that the hazard rate of X is Ax, the hazard rate of Y is ly, and the mean of Z is 4. You are also given that E (Y + Z) = Var (Y - X) and Var (X + Y + 2) = 3E (2Y + Z). Find dy - dx. Possible Answers A -0.05 D 10.05 20.09