Let X represent the amount of time students use when they take a final exam. The allotted time is up to 2 hours. Suppose X is continuous and its pdf is below.

f(x) = (3/8)x² where 0 < x < 2 (in hours)

1. Show that f(x) is a legitimate pdf (2 conditions - see lecture notes)
2. Graph f(x), including a scale for the X and Y axes.
3. Find P(X < 1).
4. Without doing another integral, what is P(X > 1)?
5. Find the MEAN amount of time students use to take the exam. Show all work.
6. Find the MEDIAN amount of time students use to take the exam. Show all work and round your answer to 3 decimal places. HINT: The total area below the median we KNOW is equal to 0.50. First set the integral equal to .50. Your job is now to figure out what your upper limit on the integral should be, in order to get the total area equal to 0.50. Just call that upper limit "M" for now, then solve for M.
7. SET UP BUT DO NOT CALCULATE the integral that finds us the variance of the amount of time students use to take the exam. Be sure to include the limits of integration.
8. To find the standard deviation, what do you do with the previous result?