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 ) = (x+6)^0.5 / C , 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 <= x ).

"Hint": To double-check your answer: should be F X ( 3 ) = 0, F X( 75 ) = 1.

1. (continued) 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 ) = 5*((2x+75)^0.5) .

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 ). "Hint": F Y ( y ) = P( Y <= y ) = P( g (X ) <= y ) = ... .

e) Use the change-of-variable technique to find the p.d.f. of Y, f Y( y ). "Hint": fY ( y ) = fX ( g - 1(y )) dx/dy . "Hint": To double-check your answer: should be f Y( y ) = F Y ' ( y )