EE 503 : HW 4 Due : 10/3/2016, Monday in class. 1. Let X1 and X2 be two random variables. Define G as G = E [max (X1 , X2 )] + E [min (X1 , X2 )] . Express G in terms of E[X1 ] and E[X2 ]. 2. A point is uniformly distributed within the disk of radius 1, i.e., its density is f (u, v) = C, 0 u2 + v 2 1 Find the probability that its distance from the origin is less than x, 0 x 1. 3. Two continuous random variables X and Y are described by the pdf k, if 0 | x || y |, 0 | y | 1 fXY (x, y) = 0, otherwise where k is a constant. a) Find k. b) Are X and Y independent? c) Are X and Y uncorrelated? (Reminder: Two random variables X and Y are uncorrelated if E(XY ) = E(X)E(Y )) 4. Cauchy-Schwartz Inequality: Prove the following inequality: [E(XY )]2 E(X 2 )E(Y 2 ) (Hint: Use the fact that for any real a, E((X + aY )2 ) 0.) 5. X and Y are two i.i.d Gaussian random variables with mean 0 and variance 2 . Find the pdf of a) R = X 2 + Y 2 b) = arctan(Y /X) Hint: X = R cos , Y = R sin 6. Let X1 , X2 , X3 and X4 be independent continuous random variables with a common density function f and let p = P (X1 < X2 < X3 < X4 ) a) Argue that the value of p is the same for all continuous density functions f . (Hint: Remember the inverse transformation method of generating random variables) b) Find p by integrating the joint density function over the appropriate region. c) Find p by using the fact that all 4! possible orderings of X1 , , X4 are equally likely. 1 \f