STAT 610-602
Fall 2014
Homework 1 Solutions
Problem 1: Prove the nite version of De Morgans Laws. Let A1 , . . . , An S. Prove that
(i) (n Ai )c = n Ac ,
i=1
i=1 i
(ii) (n Ai )c = n Ac .
i=1
i=1 i
Proof of (i):
x (n Ai )c x n Ai x Ai for any i x Ac for ev
STAT 610-602
Fall 2014
Homework 3 Solutions
Problem 1.
a) Consider f (x) = cx3 1(1,) (x) with c R. Determine c such that f is a probability density
function and nd the corresponding cumulative distribution function F .
1=
1
c/x3 dx = c/(2x2 )| = c/2 c = 2
STAT 610-602
Fall 2014
Homework 4 Solutions
Problems 1-3: numbers 2.11 (b), 2.14 (a) and 2.16 from the textbook.
2.11 (b) Y = |X|, fX (x) =
2
1 ex /2 ,
2
FY (y) = P (|X| y) = FX (y) FX (y),
1
1
2
2
fY (y) = FX (y) FX (y) = ey /2 + ey /2 =
2
2
EY =
2 y2 /2
STAT 610-602
Fall 2014
Homework 6 Solutions
Problems 1-2:
4.16 Consider independent r.v.s X, Y geometric(p).
(a) Show that U = min(X, Y ) and V = X Y are independent.
(i) V > 0 min(X, Y ) = Y :
P (U = u, V = v) = P (Y = u, X = u + v) = p(1 p)u1 p(1 p)u+v1
STAT 610-602
Fall 2014
Homework 7 Solutions
Problem 1: Independent random variables X and Y (with nite second moments) are always
uncorrelated but the reverse implication is not necessarily true. Illustrate this statement with
the following example.
0
1
2
STAT 610-602
Fall 2014
Homework 5 Solutions
Problems 1-3:
3.18 Let Y NB(r, p) and G gamma(r, 1).
p
1 (1 p)ept
MpY (t) =
holds since
r
1
1t
r
(p 0)
f (p)
p
1
=
as p 0, which can be proved with LHpital:
o
pt
g(p)
1 (1 p)e
1t
1
1
1
1
f (p)
=
= pt
0
=
.
(p)
STAT 610-602
Fall 2014
Homework 5 (due on Friday October 24)
Problems 1-3: numbers 3.18, 3.28 (c)-(e) and 3.39 from the textbook.
Problem 4: Consider a lifetime T with hazard function
h(t) = 1
et
,
(1 + et )2
t 0.
Find the cdf F (t) of T and check your a
STAT 610-602
Fall 2014
Homework 6 (due on Friday October 31)
Problems 1-2: numbers 4.16 (a) and (c), and 4.17 from the textbook.
Problem 3: Let the random variable X represent the number of successes in n independent
Bernoulli trials with success probabil
STAT 610-602
Fall 2014
Homework 4 (due on Friday October 17)
Problems 1-3: numbers 2.11 (b), 2.14 (a) and 2.16 from the textbook.
Problem 4: Let X be a random variable with range cfw_0, 1, 2, . . .. The discrete version of the
formula from problem 2.14 in
STAT 610-602
Fall 2014
Homework 2 (due on Friday September 26, 2014)
Problems 1-3: numbers 1.36, 1.39 and 1.44 from the textbook.
Problem 4. Suppose there were seven road accidents in one week. What is the probability
that they all happened on dierent day
STAT 610-602
Fall 2014
Homework 7 (due on Friday November 7)
Problem 1: Independent random variables X and Y (with nite second moments) are always
uncorrelated but the reverse implication is not necessarily true. Illustrate this statement with
the followi
STAT 610-602
Fall 2014
Homework 9 Solutions
Problem 1 (5.31)
1
X = n n Xi , n = 100, iid data with EXi = , V arXi = 2 = 9.
i=1
Apply Chebyshev:
1
0.09
1
P (|X | ) = 1 P (|X | > ) 1 2 V ar X = 1 2 2 = 1 2 .
n
Then 1 0.09 0.9 0.1 0.09 2 0.09 = 0.9 0.9 = 0.9
STAT 610-602
Fall 2014
Homework 9 (due on Monday December 1)
Problem 1: number 5.31 from the textbook.
Problem 2: a random variable T that has a t distribution with n > 2 degrees of freedom can
be represented as T = Z , where Z normal(0, 1) and Y chi-squa
STAT 610-602
Fall 2014
Homework 8 (due on Friday November 21)
Problems 1-4: numbers 4.44, 5.1, 5.21 and 5.22 from the textbook.
Problem 5: consider Sn = n iUi , where U1 , U2 , . . . are iid random variables with a coni=1
tinuous uniform distribution on (
STAT 610-602
Fall 2014
Homework 3 (due on Friday October 3, 2014)
Problem 1.
a) Consider f (x) = cx3 1(1,) (x) with c R. Determine c such that f is a probability density
function and nd the corresponding cumulative distribution function F .
b) Consider f
STAT 610-602
Fall 2014
Homework 2 Solutions
Problems 1-3:
1.36
We have 10 independent Bernoulli trials with success probabilty p = 1/5. The number of
successes X therefore has a binomial distribution, X bin(10, 1/5). This gives
1
P (X 2) = 1 P (X < 2) = 1
STAT 610-602
Fall 2014
Homework 1 (due on Friday September 19, 2014)
Problem 1: Prove the nite version of De Morgans Laws. Let A1 , . . . , An S. Prove that
(i) (n Ai )c = n Ac ,
i=1
i=1 i
(ii) (n Ai )c = n Ac .
i=1
i=1 i
Problems 2-5: numbers 1.13, 1.24,
Function of a random variable
Assume that X is a random variable with pmf/pdf fX , cdf FX .
Denote the sample space of X by X . Then any function of X ,
say Y = g(X ), is also a random variable.
Examples: Y = X , X + 5, 3X 2 , expX , |X |, Icfw_X > 0
Deno
Discrete distributions
The range (sample space) of X is countable.
Discrete Uniform
Hypergeometric
Binomial
Poisson
Geometric
Negative binomial
Huiyan Sang
Chapter 3: Common Families of Distributions
Examples
Let X = the number of insects that a spider we
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Huiyan Sang
Chapter 1: Probability Theory
What is Statistics
Statistics:
Central Limit Theorem
Let X1 , X2 , . . . be iid with mean and variance 2 . Then
n )
n(X
d N(0, 1)
.
Example: A shooter hits a target with probability p
independently in each attempt. She decides to hit the target r
times. Let X stand for the number of at
Bivariate Transformation
Consider the following bivariate transformation of (X , Y ):
U = g1 (X , Y ), V = g2 (X , Y )
Huiyan Sang
Chapter 4: Multiple Random Variables
Bivariate Transformation: Discrete Case
Assume that (X , Y ) is a discrete bivariate ra
Expected Values
Let X be a random variable with pdf or pmf f (x). The expected
value or mean of g(X ) is defined as
(R
g(x)f (x)dx if X is continuous
E(g(x) = P
xX g(x)f (x), if X is discrete
provided that the integral or sum exists.
If E|g(X )| = , we
Distribution family
The family of distributions: a class of pmfs/pdfs indexed by one
or more parameters. For example, N(, 1), Unif(a, b).
Distributions in one family have a common pdf/pmf form
but different parameter values.
For each distribution, we stud