If the joint probability density of X and Y is given by: f(x, y) = 6y      if 0 < x < 1, 0 < y < 1, x + y < 1              0        otherwise.   a)Find marginal densities of X and Y b) Are X and Y are independent? c)Find the conditional probability of Y given that X = 1/2. d) Find covariance of r.v.’s X and Y , i.e., cov(X, Y ). Comment on their correlation based on covariance value.

Please answer i,i,ii, and show steps if possible. Do not randomly write wrong answers if don't know how to solve.

2. Let X and Y be random variables, and let a and b be constants. (a) Starting from the definition of covariance, show that Cov(aX,Y) = a Cov(X,Y). You may find it helpful to remember that if EX = µx, then EaX = aux. (b) Show that Cov(X + b, Y) = Cov(X,Y). Now let X, Y, Z be independent random variables with common variance o?. (c) Find the value of Corr(2X – 3Y + 4, 2Y – Z – 1). You may use any facts about covariance from - the notes, including those from parts (a) and (b) of this question, provided you state them clearly.

The data for the joint probability mass function of X and Y (two different measurement systems) aregiven in the table below.a) Calculate the marginal distributions of X and Y and plot them.b) Select one of the Y values from the table and find the conditional probability mass function of X for thatY value you have selected and plot it.c) Show whether X and Y are independent or not.d) Calculate the covariance of (X,Y) i.e. Cov(X,Y).