Prove that any two independent random variables are uncorrelated.
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please help me! thanks 3, (20%) Prove that if any two of the three random variables X, Y, and Z are independent, I(X; Y) I(X; Y1Z) holds. 3, (20%) Prove that if any two of the three random variables X, Y, and Z are independent, I(X; Y) I(X; Y1Z) holds.
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 that Z1 and Z2 are uncorrelated random variables with zero mean and unit variance. Consider the process defined by Yt = Z1 cos(ωt) + Z2 sin(ωt) + et where et ∼ iid N(0,σ2 e) and {et} is independent of both Z1 and Z2. Prove that {Yt} is stationary.
How to prove that two independent N(0,1) random variables are the same? What's the probability of this event?
4. Let Xi, X2,... be uncorrelated random variables, such that Xn has a uniform distribution over -1/n, 1/n]. Does the sequence converge in probability? 5. Let Xi,X2 be independent random variables, such that P(X) PX--) Does the sequence X1 +X2+...+X satisfy the WLLN? Converge in probability to 0?
11. Short proofs (a) Show that if the rvs Y and Yare independent, they are uncorrelated. (b) Let X and Y be uncorrelated random variables with non-zero means. Can X and Ybe orthogonal? (c) Let X, Y and Z be random variables such that (i) Are X and y orthogonal? (Show why or why not.) (ii) Are X and y correlated? (Show why or why not.)
Prove that Box-Muller method described in class generates independent standard normal random variables. 4 a) Prove that the Box-Muller method described in class generates independent standard ll generate n to write a function which wi σ-) random variables b) Suppose that X is an exponential random variable with rate parameter λ and that Y is the integer part of X. Show that Y has a geometric distribution and use this result to give an algorithm to generate a random sample...
(e) (2 pts) Show that, if X and Y are two uncorrelated (ie. EXY = EXEY) Bernoulli (Indicator) random variables then they are independent.
EXEY) (c) (2 pts) Show that, if X and Y are two uncorrelated (i.e. EXY Bernoulli (Indicator) random variables then they are independent.
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...