Expected Value of a Function of a Random Variable Questions

1) A tool and die company makes castings for steel stress-monitoring gauges. Their annual profit, Q, in hundreds of thousands of dollars, can be expressed as a function of product demand, y:

Q(y) = 2(1-e^(-2y))

Suppose that the demand (in thousands) for their castings follows an exponential pdf, fy(y) = 6e^(-6y), y > 0. Find the company's expected profit.

2) The hypotenuse, *Y*, of an isosceles right triangle is a random variable having a uniform pdf over the interval [6,10]. Calculate the expected value of the triangle's area. Do not leave the answer as a function of *a*. (It shows the triangle with both smaller sides as *a* and the hypotenuse as *Y*.)