STAT 410
Fall 2009
Homework #3
(due Friday, September 18, by 3:00 p.m.)
1.
Consider two continuous random variables X and Y with joint p.d.f.
2 2 81 x y f X, Y ( x, y ) = 0 0 < x < K, 0 < y < K otherwise
a)
Find the value of K so that f X, Y ( x, y ) is a
STAT 410
Fall 2009
Homework #4
(due Friday, September 25, by 3:00 p.m.)
1.
Let X, Y, and Z be i.i.d. Uniform [ 0 , 1 ] random variables Find the probability distribution of W = X + Y + Z. That is, find f W (w) .
Hint: If V = X + Y, we know the p.d.f. of V
STAT 410
Examples for 09/03/2008
Fall 2008
Mixed Random Variables: 1.
Consider a random variable X with c.d.f.
0
F( x ) =
x <1
1 x < 2
x2 -2x+2
4 1
x2
a) b)
Find X = E ( X ).
2 Find X = Var ( X ).
Discrete portion of the probability distribution of X:
p (
STAT 410
Fall 2009
Homework #1
(due Friday, September 4, by 3:00 p.m.)
1.
Consider a continuous random variable X with probability density function
2 fX( x ) = 3x
0 < x <1 o.w.
0
Find the moment-generating function of X, M X ( t ).
MX( t ) = E( e t X )
STAT 410
Examples for 10/15/2008
Fall 2008
Normal (Gaussian) Distribution:
1.
Let X be normally distributed with mean and standard deviation . Find the moment-generating function of X, M X ( t ). MX( t ) = E( etX ) =
=
et x e
1 2
z 2 2
2 e ( x )
2
2
dx
e
Test 9A
AP Statistics
Name:
Directions: Work on these sheets.
Part 1: Multiple Choice. Circle the letter corresponding to the best answer.
1. Following a dramatic drop of 500 points in the Dow Jones Industrial Average in September 1998, a
poll conducted f
STAT 410
Homework #6
(due Thursday, February 28, by 5:00 p.m.)
Spring 2008
1.
One piece of PVC pipe is to be inserted inside another piece. The length of the first piece is normally distributed with mean value 20 in. and standard deviation 0.7 in
STAT 410
Homework #2
(due Friday, September 12, by 3:00 p.m.)
Fall 2008
1.
( ~ 1.9.19 )
Let X be a nonnegative continuous random variable with p.d.f. f ( x ) and c.d.f. F ( x ). Show that E( X ) =
0
(1 - F (x ) ) d x .
2.
Suppose that X follows a uniform
STAT 410
Fall 2009
Homework #2
(due Friday, September 11, by 3:00 p.m.)
1
1.
The p.d.f. of X is f X ( x ) = x How is Y distributed?
, 0 < x < 1, 0 < < . Let Y = 2 ln X.
a)
Determine the probability distribution of Y by finding the c.d.f. of Y F Y ( y ) =
STAT 410
Fall 2009
Homework #5
(due Friday, October 2, by 3:00 p.m.)
1.
Let X and Y have the joint probability density function
x+4 y f X, Y ( x, y ) = 0
Let U = X Y and V = X. Find the joint probability density function of ( U, V ), f U, V ( u, v ). Sk
STAT 410
Homework #7
(due Friday, March 7, by 3:00 p.m.)
Spring 2008
1.
Suppose X has a multivariate normal N 3 ( , ) distribution with
7 4 and covariance matrix = 2 0
mean =
17 23
2 9 - 10 . 0 - 10 25
a)
Find P ( X 1 > 10 ). X1 ~ N ( 7,
1.
Let X and Y have the joint probability density function
f X , Y ( x , y ) = 10 x y 2 ,
0 < x < y < 1,
zero otherwise.
a)
Let a > 1. Find P ( Y > a X ).
b)
1
1
Find P X > Y = .
3
2
c)
1
1
Find P Y > X = .
3
2
d)
Find E ( X | Y = y ).
e)
Find E ( Y | X =
STAT 410
Fall 2009
Homework #6
(due Thursday, October 8, by 4:00 p.m.)
1.
Suppose the size of largemouth bass in a particular lake is uniformly distributed over the interval 0 to 8 pounds. A fisherman catches (a random sample of) 5 fish. X1, X2, X3, X4, X
STAT 410
Homework #2
(due Friday, September 12, by 3:00 p.m.)
Fall 2008
1.
( ~ 1.9.19 )
Let X be a nonnegative continuous random variable with p.d.f. f ( x ) and c.d.f. F ( x ). Show that E( X ) =
0
(1 - F (x ) ) d x .
E( X ) =
0
x f (x ) d x =
0
x
0
dy
f
STAT 410
Homework #3
(due Friday, September 19, by 3:00 p.m.)
Fall 2008
1.
Suppose that the random variables X and Y have joint p.d.f. f ( x, y ) given by
f ( x, y ) = C x 2 y,
a) Sketch the support of ( X , Y ).
0 < x < y, x + y < 2.
b)
What must the val
STAT 410 Exam 1 Version A Solutions 1. The inverse transformation is x = e-y/2 , and the derivative of the inverse transformation is dx 1 = - e-y/2 . dy 2 It follows that the pdf of Y is 1 1 3 fY (y) = fX (e-y/2 ) e-y/2 = 3e-y e-y/2 = e-3y/2 2 2 2 for y >
STAT 410 Exam 1 Version B Solutions 1. (a)
1 x/2 0
P (X > 2Y ) =
0
15xy 2 dy dx =
0
1
15x
y3 3
x/2 0
1
dx =
0
5 4 5 1 1 x dx = = . 8 8 5 8
(b) fX (x) =
0
x
15xy 2 dy = 15x
0
x
y 2 dy = 15x
x3 = 5x4 3
for 0 < x < 1 (the support is (0, 1).
1
fY (y) =
y
15x
Statistics 410/Math 464
Course Syllabus
Title: Introduction to Mathematical Statistics and Probability II Catalog Description of Course: Same as Mathematics 464. continuation of statistics 400. Includes moment-generating functions, transformations of rand
Statistics 410/Math 464
Course Syllabus
Title: Introduction to Mathematical Statistics and Probability II Catalog Description of Course: Same as Mathematics 464. continuation of statistics 400. Includes moment-generating functions, transformations of rand
1. Let X and Y be continuous random variables with joint pdf
f (x, y) = 50x4 y 4
for 0 < y < 1, 0 < x < y; zero elsewhere.
(a) (10 points) Find the support and marginal pdf of Y .
The support of Y is (0, 1). The marginal pdf of Y is
fY (y) =
Z
y
0
50x4 y
a Us: .11 (I) Ind Ihe c.d.t X :1 Use art a and the cdi X
'- p / p 8 a: x
X 0 9115; https'l/wwwxoulseheracorn/iiie/p7vmn/drUserpartVarandrthchdfrapproachrtorfmdrthercdfrthatrdescribethhernengvades/ Q m x?! P
(1) Use pan (a) and the edit approach to nd the
STAT 410
Fall 2017
A. Stepanov
Examples for Week 1 (1)
Functions of One Random Variable
Example 1:
y = x2
pY( y ) = pX( y )
1
4
9
16
0.2
0.4
0.3
0.1
x
pX( x )
1
2
3
4
0.2
0.4
0.3
0.1
x
pX( x )
y
pY( y )
2
0.2
0
p X ( 0 ) = 0.4
0
0.4
4
p X ( 2 ) + p X ( 2
STAT 410
1.
Fallr 2017
A. Stepanov
Practice Problems 3
1.9.18 ( 7th edition )
1.9.17 ( 6th edition )
Find the mean and the variance of the distribution that has the cdf
F( x ) =
2.
1.9.23 ( 7th edition )
0
x0
x
0 x2
8
x2
16
1
2 x4
4 x.
1.9.22 ( 6th editi
STAT 410
1.
Fall 2017
A. Stepanov
Practice Problems 4
Let X and Y have the joint p.d.f.
f X , Y ( x, y ) = C x 2 y 3,
0 < x < 1, 0 < y <
x,
zero elsewhere.
a)
What must the value of C be so that f X , Y ( x, y ) is a valid joint p.d.f.?
b)
Find P ( X + Y