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MA22S6 Numerical and Data Analysis 1 2015-2016 Homework sheet 5 Due Thursday 10th of March 2015 in class 1. a) We consider two random variables X and Y with expectation value µX and µY respectively. Show that the covariance Cov(X, Y ) = E[(X − µX )(Y − µY )] can also be written as Cov(X, Y ) = E[XY ] − µX µY . b) Prove that that if two random variables X and Y are independent then X and Y are uncorrelated (i.e. their covariance vanishes) c) For 3 random variables X, Y, Z show that V ar[X + Y + Z] = V ar[X] + V ar[Y ] + V ar[Z] + 2Cov(X, Y ) + 2Cov(X, Z) + 2Cov(Y, Z) 2. Given the joint probability mass function (ie the probability that X = x and Y = y) pX,Y (x, y) = P (X = x, Y = y) in the following table y 1 2 3 1 0 1/4 0 x 2 1/4 0 1/4 3 0 1/4 0 Show that X and Y are dependent but uncorrelated. 3. The correlation coefficient ρ(X, Y ) is defined by ρ(X, Y ) = p Cov(X, Y ) V ar[X]V ar[Y ] provided that V ar[X] 6= 0 and V ar[Y ] 6= 0, otherwise ρ(X, Y ) = 0. a) Relate ρ(rX + s, tY + u) (with real numbers r, s, t, u) to ρ(X, Y ). b) ρ takes values in [−1, 1]. In which cases do you find ρ(X, Y ) = 1, −1, respectively? 4. A dart is thrown at a disk of radius 1 around the origin in the x − y plane. The random variables X, Y measure p the x- and y-coordinates with joint probability density function fX,Y (x, y) = c (c > 0) for x2 + y 2 ≤ 1 and fX,Y (x, y) = 0 otherwise (i.e. no dart misses the disk). a) Determine the constant c. b) Calculate the marginal probability density functions fX (x) and fY (y). c) Are X and Y independent random variables? d) Are X and Y correlated? 1 Lecturer: Stefan Sint, [email protected]