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
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 (
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
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
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
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 #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
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
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
Chapter 8 Review
AP Statistics
Name
Multiple Choice
1. A basketball player makes 70% of her free throws. She takes 7 free throws in a game. If the
shots are independent of each other, the probability that she makes 5 out of the 7 shots is
about
(a) 0.635.
Ch. 7 Review
IB Statistics
1. Suppose X is a random variable with mean . Suppose we observe X many times and
keep track of the average of the observed values. The law of large numbers says that
A) the value of will get larger and larger as we observe X.
B
Chapter 8 Review
AP Statistics
Name
Multiple Choice
1. A basketball player makes 70% of her free throws. She takes 7 free throws in a game. If the
shots are independent of each other, the probability that she makes 5 out of the 7 shots is
about
(a) 0.635.
Chapter 8 Review
AP Statistics
Name
Multiple Choice
1. A basketball player makes 70% of her free throws. She takes 7 free throws in a game. If the
shots are independent of each other, the probability that she makes 5 out of the 7 shots is
about
(a) 0.635.
Chapter 8 Review
AP Statistics
Name
Multiple Choice
1. A basketball player makes 70% of her free throws. She takes 7 free throws in a game. If the
shots are independent of each other, the probability that she makes 5 out of the 7 shots is
about
(a) 0.635.