Biostat 341
Bian Li
Homework 10
5.44
Proof
a. Given that Xi Bernoulli(p), we have
2 = p(1 p).
According to the Central Limit Theorem, n(Xn )/ N (0, 1) in distribution.
Hence, n(Yn p)/p(1p) N (0, 1) and n(Yn p) N [0, p(1p)] in distribution.
=p
and
b. Le
Biostat 341
Bian Li
Homework 6
4.1
Solution Let A be the event such that
A = cfw_(x, y) : x (a, b), y (c, d)
and the pdf for the random point be f (x, y), then
b
d
f (x, y) dxdy.
P (A) =
c
a
1
Given that f (x, y) = 4 , we have
P (A) =
d
1
4
b
c
a
1
dxdy =
Chapter 1
Basis of Probability Theory
1.1
Set Theory
Denition 1.1.1 The set, S, of all possible outcomes of a particular experiment is called the
sample space for the experiment.
If the experiment consists of tossing a coin, the sample space contains two
Review on Set Theory
Bian Li
August 29, 2015
1
Key Concepts
Experiment process of observation or measurement
Sample Space set of all possible outcomes
Event subset of the sample space
Subset A B x A x B
Union A B cfw_x | x A or x B
Intersection A B cfw_x
Biostat 341
Bian Li
Homework 9
5.29
Solution Let Xi be a random variable that describes the weight of the booklets. Then Xi s
are independent and identically distributed with = 1 and 2 = 0.0025.
P (100 booklets weigh more than 100.4 ounces) = P (X1 + + X1
Biostat 341
Bian Li
Homework 3
2.15
Proof We rst prove that
X + Y = (X Y ) + (X Y ).
Given that
X Y = min(X, Y )
X Y = max(X, Y ),
we have
(X Y ) + (X Y ) = min(X, Y ) + max(X, Y ).
Assuming X < Y without loss of generality, we have
min(X, Y ) + max(X, Y
Additional Homework Problems
Biostat 341
August 22, 2012
1 Lectures 1-2
1. Let P (A) = 1/2, P (B) = 1/8, and P (C) = 1/4, where A, B, and C are disjoint. Find
the following:
a P (A B C).
b P (AC B C C C ).
2. If P (A) = 1 and P (B C ) = 1 , can A and B be
Biostat 341
Bian Li
Homework 2
1.49
Proof By denition,
FX (t) = P (X t) and
FY (t) = P (Y t)
Therefore, given that,
FX (t) FY (t) t
and FX (t) < FY (t) for sme t
we have
P (X t) P (Y t) t and P (X t) < P (Y t) for some t
However,
P (X t) = 1 P (X > t)
and
Biostat 341
Bian Li
Homework 1
1. Let P(A) = 1 , P(B) = 1 , and P(C) = 1 , where A, B, and C are pairwise disjoint. Find the
2
8
4
following:
1. P(A B C)
2. P(AC BC CC )
Solution a. According to the Axioms of Probability, for pairwise disjoint events A1 ,
Biostat 341
Bian Li
Homework 1
1.49
Proof By denition,
FX (t) = P (X t) and
FY (t) = P (Y t)
Therefore, given that,
FX (t) FY (t) t
and FX (t) < FY (t) for sme t
we have
P (X t) P (Y t) t and P (X t) < P (Y t) for some t
However,
P (X t) = 1 P (X > t)
and