In class, we studied boa.r the probabilities a random variable X, are determined by its probability densityr function f[$) More precisely, if X takes values in [14, B], then x} is a function on [14, B], satisfying 1. x} 2 G on [441,3] 2. ,rf mm = 1 3. If [a,b} is a subinterval of [14,3], then P(a. g X <_z b i: f lnformallyr probabilites are given by areas under the curve y="r)" in this project we will consider two random variables x and probabilities that they take certain pairs of values these determined a probability density function joint more precisely if takes then is on rectangle r="[441," satisfying at. d r. ffrf subregion phx ila-fl :1:_ informally volumes surface z .15 any rst condition called mara-e x... g another variable expected value double integral wxnpffa my mom-y for example were trying to determine sun would compute elx gamma company process designing new product. there components required build their each costs between zero one dollar. unfortunater precise cost random. let be representing component two. el ya verify indeed greater than or equal hint: draw appropriate region a. solution: after updating design decides unit product require units how much should expect it make solution>