Question 1 、 Let X, Y and Z be three random variables that take values in the alphabet {0,1, M-l...
Let X and Y be iid uniform random variables on [0,1]. Find the pdf of Z=X+Y
Problem 3 Let X be Uniform(0,1) and Y be Exponential (1). Assume that X and Y are independent. i. Find the PDF of Z- X +Y using convolution. ii. Find the moment generating function, øz(s), of Z. Assume that s< 0. iii. Check that the moment generating function of Z is the product of the moment gen erating functions of X and Y Problem 3 Let X be Uniform(0,1) and Y be Exponential (1). Assume that X and Y are...
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, Y, Z be independent uniform random variables on [0,1]. What is the probability that Y lies between X and Z.
. Let Y and Z be independent uniform random variables on the interval [0,1]. Let X = ZY. (a) Compute E(XY). (b) Compute E(X).
Let the random variable X have a uniform distribution on [0,1] and the random variable Y (independent of X) have a uniform distribution on [0,2]. Find P[XY<1].
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...
4. Let Y and Z be independent uniform random variables on the interval [0,1]. Let X Z (a) Compute E(XTY). (b) Compute E(X).
The numbers x, y, z are independent with uniform distribution on [0,1]. Find the probability that one can construct a triangle with sides length x, y, z.
12. Let X and Y be independent random variables, where X has a uniform distribution on the interval (0,1/2), and Y has an exponential distribution with parameter A= 1. (Remember to justify all of your answers.) (a) What is the joint distribution of X and Y? (b) What is P{(X > 0.25) U (Y> 0.25)}? nd (c) What is the conditional distribution of X, given that Y =3? ur worl mple with oumbers vour nal to complet the ovaluato all...