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 l. A company is in the process of designing a new product. There are two components required to build their new product. Each of these components costs between zero and one dollar. Unfortunately the precise cost of each components is random. Let X be the random variable representing the cost of component one, and Y be the random variable representing cost of component two. The joint probability density function is f (3:, y) : Eh + 3/2). Verify that this is indeed a joint probability density function, and determine the probability that the cost of the component one is greater than or equal to the cost of component two. Hint: Draw the appropriate region A. Solution