Let X, Y, Z be random variables. Prove or disprove the following statements. (That means, you need to either write down...
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...
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.
Write a formal proof to prove the following conjecture to be true or false. If the statement is true, write a formal proof of it. If the statement is false, provide a counterexample and a slightly modified statement that is true and write a formal proof of your new statement. Conjecture: 15. (12 pts) Let h: R + RxR be the function given by h(x) = (x²,6x + 1) (a) Determine if h is an injection. If yes, prove it....
5. Let X, Y, Z be random variables with joint density (discrete or continuous) plr, y,a) a f(x, 2)g(y, 2)h() Show that (a) p(rly, s) x /(r, :), ie. P(rly, :) is a function of 1 and :; (b) p(y|z, z) g(y, z), İ.e. p(y|z,z) is a function of y and z; (c) X and Y are conditionally independent given Z
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)...
1. Fundamentals: (a) Briefly, state why probability is important for statisticians (b) Let random variables X, Y, and Z be distributed according to the following table. probability 1/4 1/4 i. True or false: X and Y are independent. Explain. ii. True or false: X and Y are conditionally independent given Z. Explain. (c) Let A, B, and D be events, where 0< PD) 1. i. Prove that P(An B P(AB) 2 P(A) +P(B) 1. ii. Suppose that P(AD) 2 P(B|D)...
Random variables z and y described by the PDF if x-+ yo 1 and x.> 0 and y, > 0 0 otherwise a Are x and y independent random variables? b Are they conditionally independent given max(x,y) S 0.5? c Determine the expected value of random variabler, defined byr xy.
33. Let X and Y be independent exponential random variables with respective rates λ and μ. (a) Argue that, conditional on X> Y, the random variables min(X, Y) and X -Y are independent. (b) Use part (a) to conclude that for any positive constant c E[min(X, Y)IX > Y + c] = E[min(X, Y)|X > Y] = E[min(X, Y)] = λ+p (c) Give a verbal explanation of why min(X, Y) and X - Y are (unconditionally) independent. 33. Let X...
14. Let X and Y be random variables with joint pdf for mig). { if -4exal and otherwise 2 ay 44 o (a) Find constant K. (b.) rove or desprove: X and Y are orthogonal (c) Prove or disprove: X and Y are independent.
Consider two random variables, X and Y. Let E(X) and E(Y) denote the population means of X and Y respectively. Further, let Var(X) and Var(Y) denote the population variances of X and Y. Consider another random variable that is a linear combination of X and Y Z- 3X- Y What is the population variance of Z? Assume that X and Y are independent, which is to say that their covariance is zero.