duction to Stochastic Process, Assigment 1 (15%) Due day: Feb 22, 2016, due in class 0.1. Suppose that the joint probability density function of X and Y is dened as follows: f (x, y) = x + y, for 0 x 1, and 0 y 1, otherwisef (x, y) = 0. (1) Find E(Y |X) and E(X|Y ). 0.2. Suppose that three random variable X1 , X2 , and X3 have a continous joint distribution function with the following joint probability density function: f (x1 , x2 , x3 ) = c(x1 + 2x2 + 3x3 ) for 0 xi 1(i = 1, 2, 3), (2) otherwise it is 0. 1. Determine the value of constant c, 2. Determine the marginal joint probability density distribution of X2 and X3 3. Determine Pr(X3 < 1 2 1 3 | X1 = 4 , X2 = 4 ) 0.3. A club has 120 members, of whom 35 play chess, 58 play bridge, and 27 play both chess and bridge. If a member of the club is randomly chosen, what is the conditional probability that she 1. plays chess given she plays bridge; 2. plays bridge given that she plays chess? 0.4. A lawyer must decide whether to charge a xed fee of $5,000 or take a contigency fee of $25,000 if she wins the case (and 0 if she loses). She estimates that her probability of wining is 0.3. Determine the mean and standard deviation of her fee if 1. she takes the xed fee; 2. she takes the contigency fee. 0.5. The probability density function (PDF) of a sum of two continous random variables X and Y is given by the convolution of the PDFs, fX and fY : fX (x)fY (z x)dx fX+Y (z) = 1 (3) 1. Show that a sum of two independent standard normal variables results in a normal variable. Find the PDF of such a sum. 2 2 2. Assuming that X1 N orm(1 , 1 ) and X2 N orm(2 , 2 ) are correlated with Corr(X1 , X2 ) = , nd the mean and variance of a1 X1 + a2 X2 , for a1 , a2 R. 2