Problem 11

Melvin Fooch, a former student in ECE 460, has found that the hours he spends working (W) and sleeping (S) in preparation for the quiz are random variables having the joint density

$$f_{W,S}(w,s) = \left\{ \begin{array} {ll} K & \mbox{if $10 ... ...w$, $0 \leq s$} \\ 0 & \mbox{otherwise} \end{array} \right.$$

What Melvin does not know, and even his best friends will not tell him, is that working only furthers his confusion and that his grade G is given by

G = 2.5(S-W) + 50.

1.

Determine the constant K.

2.

What is the probability that the maximum time he devotes to either working or sleeping is less than 10 hours? (In other words, what is the probability that S <10 and W<10?)

3.

The instructor will pass poor Melvin if he achieves a grade of 75 or greater. What is the probability that this will occur?

4.

Find the density function of the random variable G.

Hint: You may find this problem easier if you first plot the range of s and w where f_SW(s,w) is non-zero and then use the appropriate areas in that plot to compute your integrals.