(e) (2 pts) Show that, if X and Y are two uncorrelated (ie. EXY = EXEY)...
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 2. (6 pts) Independence and Conditional Probability (a) (2 pts) An urn contains 3 red and 5 green balls. At each step of this game, we pick one ball at random, note its color and return the ball to the urn together with anoter ball of the same color. Prove by induction that the probability that the ball we pick a red ball at the n-th step is 3/8. (b) (2pts) Consider any two random variables X, Y of...
(a) (2 pts) An urn contains 3 red and 5 green balls. At each step of this game, we pick one ball at random, note its color and return the ball to the urn together with anoter ball of the same color. Prove by induction that the probability that the ball we pick a red ball at the n-th step is 3/8. (b) (2pts) Consider any two random variables X, Y of any distirbution and not necesarily independent. Given that...
X and Y are random variables (a) Show that E(X)=E(B(X|Y)). (b) If P((X x, Y ) P((X x})P({Y y)) then show that E(XY) = E(X)E(Y), i.e. if two random variables are independent, then show that they are uncorrelated. Is the reverse true? Prove or disprove (c) The moment generating function of a random variable Z is defined as ΨΖφ : Eez) Now if X and Y are independent random variables then show that Also, if ΨΧ(t)-(λ- (d) Show the conditional...
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.)
15 PTS) 2. Use implicit differentiation to show that x + y +exy = 0 is an implicit solution to the nonlinear equation (1 + xexy)*%+1+ yexy = 0. dx
5. (2 points) Let X and Y be Bernoulli random variables. Show that X and Y are independent if and only if Cov(X, Y) = 0.
Let X and Y be two independent random variables. Show that Cov (X, XY) = E(Y) Var(X).
a) b) YES or NO? (This may be a little tricky.) If X and Y are uncorrelated Bernoulli(p) random variables, they necessarily independent? are TRUE or FALSE? A Poisson process has stationary and independent increments; and, moreover, it is highly unlikely that two "arrivals" for such a process will occur within a very short time internal
4. Suppose that X and Y are random variables with E(X) = 2, E(Y) = 1. E(X*) = 5, E(Y2-10, and E(XY) = 1 (a) Compute Corr(X,Y) (b) Choose a number c so that X and X +cY are uncorrelated