STAT 410
Spring 2026 Name ANSWERS .
Version C NetID ( e-mail ) ____________________
1 2 3 4
Exam 1 TuTh
2:00 pm
MWF
9:00 am
MWF
10:00 am
TuTh
9:30 am
Douglas Stepanov Stepanov Douglas
Page Earned Please circle your section.
1 The exam has 4 problems and 9 pages.
2 Be sure to show all your work; your
partial credit might depend on it.
3 Put your final answers at the end of
your work and mark them clearly.
4 Box the final answers where appropriate.
If the answer is a function,
5 specify what kind of function
it is ( p.d.f. or c.d.f. ), and
6 its support must be included.
7 No credit will be given
8 without supporting work.
The exam is closed book and closed notes.
9 You are allowed to use a calculator and
one 8½" x 11" sheet (both sides) with notes.
Total / 80 Turn in all scratch paper with your exam.
___________________________________________________________________________
Academic Integrity
The University statement on your obligation to maintain academic integrity is:
If you engage in an act of academic dishonesty, you become liable to severe disciplinary action. Such acts include cheating;
falsification or invention of information or citation in an academic endeavor; helping or attempting to help others commit
academic infractions; plagiarism; offering bribes, favors, or threats; academic interference; computer related infractions; and
failure to comply with research regulations.
Article 1, Part 4 of the Student Code gives complete details of rules governing academic integrity for all students. You are
responsible for knowing and abiding by these rules.
, 1. (8) Let X be a continuous random variable with probability density function
x3
fX( x ) = , 0 ≤ x ≤ 4, zero elsewhere.
64
1
Find the probability distribution of Y = .
X
1 1 1
0<x<4 0< x <2 > > . y> .
x 2 2
x u4 x
u3 x4
FX( x ) = P( X x ) = 64
du
0
, 0 ≤ x < 4.
0 256 256
1 1
FY( y ) = P( Y ≤ y) = P( ≤ y) = P( X )
X y2
1 1 1
= 1 – FX( ) = 1– , y .
y2 256 y 8 2
1
F Y ( y ) = 0, y< .
2
OR
1 1 –1 dx 2
y= x= =g (y) =
x y2 dy y3
dx 1 2 1 1
f Y ( y ) = f X ( g – 1( y ) ) = = , y> .
dy 64 y 6 y3 32 y 9 2
d 1 1
Indeed,
dy
1
256 y 8
=
32 y 9
.
Spring 2026 Name ANSWERS .
Version C NetID ( e-mail ) ____________________
1 2 3 4
Exam 1 TuTh
2:00 pm
MWF
9:00 am
MWF
10:00 am
TuTh
9:30 am
Douglas Stepanov Stepanov Douglas
Page Earned Please circle your section.
1 The exam has 4 problems and 9 pages.
2 Be sure to show all your work; your
partial credit might depend on it.
3 Put your final answers at the end of
your work and mark them clearly.
4 Box the final answers where appropriate.
If the answer is a function,
5 specify what kind of function
it is ( p.d.f. or c.d.f. ), and
6 its support must be included.
7 No credit will be given
8 without supporting work.
The exam is closed book and closed notes.
9 You are allowed to use a calculator and
one 8½" x 11" sheet (both sides) with notes.
Total / 80 Turn in all scratch paper with your exam.
___________________________________________________________________________
Academic Integrity
The University statement on your obligation to maintain academic integrity is:
If you engage in an act of academic dishonesty, you become liable to severe disciplinary action. Such acts include cheating;
falsification or invention of information or citation in an academic endeavor; helping or attempting to help others commit
academic infractions; plagiarism; offering bribes, favors, or threats; academic interference; computer related infractions; and
failure to comply with research regulations.
Article 1, Part 4 of the Student Code gives complete details of rules governing academic integrity for all students. You are
responsible for knowing and abiding by these rules.
, 1. (8) Let X be a continuous random variable with probability density function
x3
fX( x ) = , 0 ≤ x ≤ 4, zero elsewhere.
64
1
Find the probability distribution of Y = .
X
1 1 1
0<x<4 0< x <2 > > . y> .
x 2 2
x u4 x
u3 x4
FX( x ) = P( X x ) = 64
du
0
, 0 ≤ x < 4.
0 256 256
1 1
FY( y ) = P( Y ≤ y) = P( ≤ y) = P( X )
X y2
1 1 1
= 1 – FX( ) = 1– , y .
y2 256 y 8 2
1
F Y ( y ) = 0, y< .
2
OR
1 1 –1 dx 2
y= x= =g (y) =
x y2 dy y3
dx 1 2 1 1
f Y ( y ) = f X ( g – 1( y ) ) = = , y> .
dy 64 y 6 y3 32 y 9 2
d 1 1
Indeed,
dy
1
256 y 8
=
32 y 9
.
