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11. Short proofs (a) Show that if the rvs Y and Yare independent, they are uncorrelated....
Which of the following statements are true and false? Prove the true ones and give counterexamples for the false ones. Let X and Z be random variables. (i) If X and Z are uncorrelated, then they are independent. (ii) If X and Z are independent, then E[X2]
Which of the following statements are true and false? Prove the trueones and give counterexamples for the false ones. Let X and Z be random variables.(i) If X and Z are uncorrelated, then they are independent.(ii) If X and Z are independent, then E[X2] = E[Z2].(iii) If X and Z are correlated, they are also dependent.
Suppose X, Y and Z are three different random variables. Let X obey Bernoulli Distribution. The...
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...
EXEY) (c) (2 pts) Show that, if X and Y are two uncorrelated (i.e. EXY Bernoulli...
EXEY) (c) (2 pts) Show that, if X and Y are two uncorrelated (i.e. EXY Bernoulli (Indicator) random variables then they are independent.
(e) (2 pts) Show that, if X and Y are two uncorrelated (ie. EXY = EXEY)...
(e) (2 pts) Show that, if X and Y are two uncorrelated (ie. EXY = EXEY) Bernoulli (Indicator) random variables then they are independent.
1 Expectation, Co-variance and Independence [25pts] Suppose X, Y and Z are three different random variables....
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)...
In the following questions, let Bt denote a Brownian motion with Bo = 0. (a) Show...
In the following questions, let Bt denote a Brownian motion with Bo = 0. (a) Show that if random variables X and Y are independent then they are 1. uncorrelated, Cov(X, Y) -0. (b) Let X have distribution P(X-1)- P(X 0) P(X- -1)-1/3, and Y-İX . Show that X and Y are uncorrelated, but not independent. (c) Let (X, Y) be a Gaussian vector. Show that if X and Y are uncorrelated then they are independent.
Problem 1: Random variables Y, and Y, are uncorrelated. We want a linear minimum mean- square...
Problem 1: Random variables Y, and Y, are uncorrelated. We want a linear minimum mean- square error (MMSE) non-homogeneous estimate X, of the value of random variable X in terms of Y, and Y" The estimate has the form XL =g(Υ.Υ,) = a1+ bY, + c . Find the values of a, b and c that minimize the expected value of the error given by ECX-+by, +c)'). Express your answer in terms of the means and variances of Y, and...
12. (a) Show that ifXandy are independent, then EC-P) = Eal) and X = sin Θ,...
12. (a) Show that ifXandy are independent, then EC-P) = Eal) and X = sin Θ, X cose. Are X and y are correlated or 0 otherwise uncorrelated? (Show why.) (c) Are the X and Y of part (b) independent? (Show why.) Hints: part (a); trig identity 2sincosin(20); indefinite integral [sin'udu -^u-sin 2u+C. c e
Let X and Y be independent exponential(1) RVs (f(x) e 10). Show that uniform(0, 1) distribution....
Let X and Y be independent exponential(1) RVs (f(x) e 10). Show that uniform(0, 1) distribution. Hint: consider defining the auxiliary X/(X Y) has a RV XY [12
Problem 5 Let X and Y be random variables with joint PDF Px.y. Let ZX2Y2 and tan-1 (Y/X). Θ i. Find the joint PDF of Z and Θ in terms of the joint PDF of X and Y ii. Find the joint PDF of Z and Θ if...
Problem 5 Let X and Y be random variables with joint PDF Px.y. Let ZX2Y2 and tan-1 (Y/X). Θ i. Find the joint PDF of Z and Θ in terms of the joint PDF of X and Y ii. Find the joint PDF of Z and Θ if X and Y are independent standard normal random variables. What kind of random variables are Z and Θ? Are they independent?
Problem 5 Let X and Y be random variables with joint...
EXEY) (c) (2 pts) Show that, if X and Y are two uncorrelated (i.e. EXY Bernoulli (Indicator) random variables then they are independent.
(e) (2 pts) Show that, if X and Y are two uncorrelated (ie. EXY = EXEY) Bernoulli (Indicator) random variables then they are independent.
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)...
In the following questions, let Bt denote a Brownian motion with Bo = 0. (a) Show that if random variables X and Y are independent then they are 1. uncorrelated, Cov(X, Y) -0. (b) Let X have distribution P(X-1)- P(X 0) P(X- -1)-1/3, and Y-İX . Show that X and Y are uncorrelated, but not independent. (c) Let (X, Y) be a Gaussian vector. Show that if X and Y are uncorrelated then they are independent.
Problem 1: Random variables Y, and Y, are uncorrelated. We want a linear minimum mean- square error (MMSE) non-homogeneous estimate X, of the value of random variable X in terms of Y, and Y" The estimate has the form XL =g(Υ.Υ,) = a1+ bY, + c . Find the values of a, b and c that minimize the expected value of the error given by ECX-+by, +c)'). Express your answer in terms of the means and variances of Y, and...
12. (a) Show that ifXandy are independent, then EC-P) = Eal) and X = sin Θ, X cose. Are X and y are correlated or 0 otherwise uncorrelated? (Show why.) (c) Are the X and Y of part (b) independent? (Show why.) Hints: part (a); trig identity 2sincosin(20); indefinite integral [sin'udu -^u-sin 2u+C. c e
Let X and Y be independent exponential(1) RVs (f(x) e 10). Show that uniform(0, 1) distribution. Hint: consider defining the auxiliary X/(X Y) has a RV XY [12
Problem 5 Let X and Y be random variables with joint PDF Px.y. Let ZX2Y2 and tan-1 (Y/X). Θ i. Find the joint PDF of Z and Θ in terms of the joint PDF of X and Y ii. Find the joint PDF of Z and Θ if X and Y are independent standard normal random variables. What kind of random variables are Z and Θ? Are they independent?
Problem 5 Let X and Y be random variables with joint...
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