Question

There are two independent Bernoulli random variables, U and V , both with probability of success 1/2. Let X=U+V and Y =|U−V|.
1) Calculate the covariance of X and Y

2) Explain whether X and Y are independent or not

3) Identify the random variable expressed as the conditional expectation of Y given X, i.e., E[Y |X].

Answer 1

(a) The values of Bernouli random Variables. U and v are independent here, the values of u and v are obtained as: Here, P(UO. VEO) - 0.5 0.5 = 0.25 POUI, V.1) - 05 X0.5 -0.25 Plast, V=o) o 5x05 -025 Puzo, V21 : 0.5 X05 - 0.25 The values of u, v are taken randomly variables considered as o and 1 Now. We taken as x,y here is obtained as: PCX20.Y-0) - PCU:0, V20) - 0-25 - 0.25 P(x=2 Y=0) P(U21, V-1) PCX21, 4:1) PCURL, V=0) + PU 20. VEI) - 0.25 +0.25 +0.5 .: The expected values are considered as E(xy) = 1X PCX=14=1) = 05 E(X) = 2X0 25 + 1X0.25 - 1 ECY) = 1X0.5 = 0.5 The covariance (x,y) = E(XY) - FCX) ECY) = 0.5 - (05) = 05 -0.5 Here we get the value of covariance (x,y) =0 The value of 0 is the required covariance.

b) Here we are using the value o for x and y When we get PCX =0) = 0 25 PCY=0) = 0.25 +0.25 = 0.5 P(x=0,4 =0) = 0.25 The value is not equal to to Px = 0) P(V):o) from the above process we are decided x and y are not independent variables c) Given x, the conditional expected value of Y here E(Y/2 = o) :o When a=0, y has to be o E(Y/x = 1) = 1 When a = 1, y has to be 1. E(Y/2 = 2)=0 When x=2 y has to be o