In class, we studied how the probabilities a random variable X, are determined by its probability density function f (as) More precisely, if X takes values in [A, B], then f (as) is a function on [A, B], satisfying 1. f(3:) 2 0 on [A,B]. 2. ff f(:c)d:r: : l 3. If [a,b] is a sub-interval of [A, B], then P(a S X S b) : f:f(a:)da:.1nformally probabilites are given by areas under the curve 3; : f(:c) In this project we will consider two random variables X and Y, and the probabilities that they take certain pairs of values (a, 3)). These probabilities are determined by a 2-Variable probability density function f (x, 3;) (called a joint probability density function). More precisely, if X takes values in [A, B], and Y takes values in [0, D], then ag) is a function on the rectangle R : [A, B] x [0, D] satisfying 1. f(:c,y) 2 0 on R. 2. ransom : 1 3. If A is a sub-region of R, then P((X, Y) is in A): f f A f (.13, y)d:cdy. Informally, probabilities are given by volumes under the surface 2 = f (I, y). Any function satisfying the rst two condition is called a joint probability density function on R. Mfume P((K,YJ a' in. A) If g(:c,y) is another two variable function then, the expected value of g(X,Y) is given by the double integral 1E[9(X,Y)]: [ngawwiwdxdy For example if we were trying to determine the expected value of the sun X + Y, we would compute IE[X + Y] : jjcc + mammary 2. After updating their design, the company decides that each unit of their new product will require two units of component one and 3 units of component two. How much should they expect it to cost to make 1000 units of their new product? Solution