In this project we will consider two random variables X and Y, and the probabilities that the},r take certain pairs of values ((1,3)). These probabilities are determined by a 2-miable probability density function f (3:, 3;) (called a joint probability density function). More precisely, if X takes values in [11,13], and Y takes values in [0,13], then f(:r,y) is a function on the rectangle R 2 IA, B] x [0, D] satisfying 1. f(;r,y] 2 O on R. 2. ffR f(;r,y)da:dy = 1 3. If A is a sub-region of R, then P((X,Y} is in A}: ffA f(:r:,y)d:rdy. Informally, probabilities are given by volumes under the surface 2: = f (I, y). An}r function satisfying the rst two condition is called a joint probability density function on R. \"We a: f((K,YJ .5 in A) If g(;r,'y) is another two variable function then, the expected value of g{X,Y) is given by the double integral ELM, m = [A screams; For example if we were trying to determine the expected value of the sun X + Y, we would compute EIX + Y1= jftz + Mammy