Coursework question 12.1 Give examples of non-trivial random variables X, Y such that E「XX21 -X and...
Let X and Y be two independent and identically distributed random variables with expected value 1 and variance 2.56. First, find a non-trivial upper bound for P(|X + Y − 2| ≥ 1). Now suppose that X and Y are independent and identically distributed N(1,2.56) random variables. What is P(|X + Y − 2| ≥ 1) exactly? Why is the upper bound first obtained so different from the exact probability obtained?
Question 19 Consider two random variables X and Y with E(X)= 4, E(Y) = 2, E(XY) = 12, V(X) = 16 and V(Y) = 25, then the correlation coefficient between X and Y is: a. -0.2 b. -0.3 c. 0.2 d. 0.3 e. None of the above need step by step distribution~
(3) Suppose X and Y are discrete random variables. Show that E(X|Y) = E(X)Y*) (3) Suppose X and Y are discrete random variables. Show that E(X|Y) = E(X)Y*)
Two random variables X and Y have means E[X] = 1 and E[Y] = 0, variances 0x2 = 9 and Oy2 = 4, and a correlation coefficient xx =0.6. New random variables are defined by V = -2X + Y W = 2X + 2Y Find the means of V and W Find the variances of V and W defined in question 3 Find Rww for the variables V and W defined in question 3
C5 a. Give a non-trivial example of an ideal in the ring Zc [x]. b. Is this a prime ideal? C. Is it a maximal ideal?
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...
Let X and Y be two independent random variables such that E(X) = E(Y) = u but og and Oy are unequal. We define another random variable Z as the weighted average of the random variables X and Y, as Z = 0X + (1 - 0)Y where 0 is a scalar and 0 = 0 < 1. 1. Find the expected value of Z , E(Z), as a function of u . 2. Find in terms of Oy and...
Question 3 [17 marks] The random variables X and Y are continuous, with joint pdf 0 y otherwise ce fxx (,y) a) Show that cye fr (y) otherwise and hence that c = 1. What is this pdf called? (b) Compute E (Y) and var Y; (c) Show that { > 0 fx (a) e otherwise (d) Are X and Y independent? Give reasons; (e) Show that 1 E(XIY 2 and hence show that E (XY) =. Question 3 [17...
For the random variables X and Y having E(X) = 1, E(Y) = 2, Var (X) = 6, Var (Y) = 9, and Pxy = -2/3. Find a) The covariance of X and Y. b) The correlation of X and Y. c) E(X2) and E(Y2).
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