question archive X and Y are two random variables whose joint probability density function (pdf ) is given by PX,Y (3, y) = Bx, (x, y) EA 0 , otherwise 3 where 3 is a normalization constant and A is a set shown in Fig
X and Y are two random variables whose joint probability density function (pdf ) is given by PX,Y (3, y) = Bx, (x, y) EA 0 , otherwise 3 where 3 is a normalization constant and A is a set shown in Fig
Subject:MathPrice: Bought3
X and Y are two random variables whose joint probability density function (pdf ) is given by PX,Y (3, y) = Bx, (x, y) EA 0 , otherwise 3 where 3 is a normalization constant and A is a set shown in Fig. 1. a.) Find B. b.) Find and sketch the marginal pdf's px(x) and py (y), respectively. c.) Find the conditional pdf pxy(xly). Are X and Y independent? d.) Find the probability P[(X, Y) E B], where B is the set shown in Fig. 1. e.) Sketch the conditional pdf pxjy(xly) for y = 1/4. Sketch it also for x = 1/4. 1 B A X X Figure 1: For Problem 4.
(1-4 points) Consider two independent random variables: x, which is A(0, 1) and m Bernoulli, with sample space {-1, 1} and equally likely outcomes. (a) Show that y = m - x is Gaussian. (b) Show that z - x + y is not Gaussian. Should this not be Gaussian as Quiz 1 stated? The answer of the next question should clarify why that is not the case. (c) Show that x and y are not jointly Gaussian. (d) Show that if x and y are jointly Gaussian then z is Gaussian (marginal Gaussian is not enough! Also if the marginal are Gaussian the example shows the joint distribution does not have to be Gaussian).
