1. Grades on Fall 2020 STAT 410 Exam 1 were not very good*. Graphed, their distribution had a shape similar to the probability density function

fX ( x ) = sqrt (x+6)/C, x + , 3 x 75, zero elsewhere.

a) Find the value of C that makes fX( x ) a valid probability density function.

b) Find the cumulative distribution function of X, F X ( x) = P( X smaller or equal to x ). As a way of "curving" the results, the instructor announced that he would replace each person's grade, X, with a new grade, Y = g (X ), where g (x ) = 5sqrt(2x+75).

c) Find the support (the range of possible values) of the probability distribution of Y.

d) Use part (b) and the c.d.f. approach to find the c.d.f. of Y, F Y( y ).

e) Use the change-of-variable technique to find the p.d.f. of Y, f Y( y ).