1. Grades on Fall 2021 STAT 410 Exam I were not very good *. Graphed, their distribution had a shape similar to the probability density function **
fx(x) 80, zero elsewhere. c
a) Find the value of C that makes A (x) a valid probability density function.
b) Find the cumulative distribution function of X, F x(x) = P (X x).
"Hint": To double-check your answer: should be Fx(80)
* The probability distribution is fictional, the exam has not happened yet. Hopefully, the actual grades will be slightly better than these.
** Exam scores should have a discrete (instead of continuous) nature. A continuous probability distribution is used as an approximation, since the alternative would have been dealing with a discrete random variable with 65 possible values ( 16, 17, 18, 79, 80 ), which is not nearly as much fun as I am describing it here.
1. (continued)
As a way of "curving" the results, the instructor announced that he would replace each person's grade, X, with anew grade, Y where g (x) = 8 x+20
c) Find the support ( the range of possible values ) of the probability distribution of Y.
d) Use pan (b) and the c.d.f. approach to find the c.d.f. of Y, Fy(y).
"Hint": Fy(y) =
e) Use the change-of-variable technique to find the p.d.f. of Y, (y ).
"Hint": fy(y) 1 0')) —dx
dy
"Hint": To double-check your answer: should be fy(y) = FY(y).
f) Is Ply equal to g(gx)?
2. Consider a continuous random variable X with the probability density function
fx(x) zero elsewhere.
18
16
Consider Find the probability distribution of Y.
2
x
You are welcome to use a computer and/or calculator on any problem to evaluate any integral. For the supporting work, you should include the full integral (with the function and the bounds) and the answer. For example,
x 3 4
J u 2 du — x x 2 y dy dx = 32, dx dy 92
3
0 0
