Math 2p81
Lecture 5
Wai Kong (John) Yuen
[email protected]
Department of Mathematics, Brock University
September 23, 2013
Wai Kong (John) Yuen [email protected]
Math 2p81
Event-composition method
With the additional laws and properties, we can use another
app
Math 2p81
Lecture 9
Wai Kong (John) Yuen
[email protected]
Department of Mathematics, Brock University
October 3, 2013
Wai Kong (John) Yuen [email protected]
Math 2p81
Example 3.10
Survey 20 workers, 6 favoured new policy. Estimate true
proportion p favouring
Math 2p81
Lecture 10
Wai Kong (John) Yuen
[email protected]
Department of Mathematics, Brock University
October 3, 2013
Wai Kong (John) Yuen [email protected]
Math 2p81
Exercise 3.90
Testing employees one by one. 40% are positive. Need to
send 3 positive empl
Math 2p81
Lecture 8
Wai Kong (John) Yuen
[email protected]
Department of Mathematics, Brock University
September 26, 2013
Wai Kong (John) Yuen [email protected]
Math 2p81
Some computational results
In Thm 3.3 - 3.6, we assume that Y is a discrete rv, c is a
c
Math 2p81
Lecture 6
Wai Kong (John) Yuen
[email protected]
Department of Mathematics, Brock University
September 20, 2013
Wai Kong (John) Yuen [email protected]
Math 2p81
Partition
In the box-ball example, we calculated P(W ) by cutting the
sample space based
1.
(a)
Pr A B C = 1 Pr(A B C) =
1 Pr(A B) + Pr(A B C) =
1 Pr(A) + Pr(A B) + Pr(A C) Pr(A B C) =
1 .5 + .21 + .25 .12 = 0.84
(b)
Pr A (B C) = Pr(B C) Pr[(B C) A] =
Pr(B) + Pr(C) Pr(B C) Pr[(A B) (A C)] =
Pr(B) + Pr(C) Pr(B C) Pr(A B) Pr(A C) + Pr(A B C)
1.
(a)
Pr A B A C = 1 Pr [(A B) (A C)] =
1 Pr(A B) Pr(A C) + Pr(A B C) = 1 0.21 0.25 + 0.12 = 66%
(b)
Pr A B A C = 1 Pr [A B A C] = 1 Pr [A B C] =
1 Pr(A) Pr(B) Pr(C) + Pr(A B) + Pr(A C) + Pr(B C) Pr(A B C) =
1 0.50 0.54 0.53 + 0.21 + 0.25 + 0.30 0.12 = 7
1.
Pr (A B C) (A B C) = 1 Pr (A B C) (A B C)
1 Pr(A B C) Pr(A B C) = 1 Pr(A B) + Pr(A B C) Pr(A C) + Pr(A B C)
= 1 Pr(B) + Pr(A B) + Pr(B C) Pr(A B C) Pr(A C) + Pr(A B C)
= 1 Pr(B) + Pr(A B) + Pr(B C) Pr(A C) = 1 0.54 + 0.21 + 0.30 0.25 = 72%
2.
Pr A (B C
1.
Pr[(A B) B C (C D)] = 1 Pr[(A B) B C (C D)
= 1 Pr[B C D)
as A B is a subset of B, and C (C D) = (C D) = C D. Now
1 Pr[B C D) = Pr(B C D) = 0.79 0.17 0.45 = 6.0435%
2.
X
Y
0
1
2
a. c =
1
40
0
1c
2c
3c
6c
1
3c
4c
0
7c
2
5c
6c
0
11c
3
7c
0
0
7c
4
9c
0
0
1.
Pr (A B C) (A B C) (A B C) = P (A B C) +
P (A B C) + P (A B C) = P (A B) P (A B C) + P (B C)
P (A B C) + P (A C) P (A B C) = P (A) P (A B) +
P (B C) P (A B C) = 0.31 0.11 + 0.12 0.05 = 27%
2.
Pr (A B C D) (A B C) (A B D) =
0.47 0.79 0.17 0.55 + 0
Math 2p81
Lecture 7
Wai Kong (John) Yuen
[email protected]
Department of Mathematics, Brock University
September 26, 2013
Wai Kong (John) Yuen [email protected]
Math 2p81
Random variables
While experimental outcomes can be anything, we are usually
interested
Math 2p81
Lecture 11
Wai Kong (John) Yuen
[email protected]
Department of Mathematics, Brock University
October 7, 2013
Wai Kong (John) Yuen [email protected]
Math 2p81
Exercise 3.103, 3.106
10 machines, 4 defective. Randomly select 5 machines. Find
probabili
Math 2p81
Lecture 4
Wai Kong (John) Yuen
[email protected]
Department of Mathematics, Brock University
September 15, 2013
Wai Kong (John) Yuen [email protected]
Math 2p81
Conditional probability - an example
Roll a die once. Let A be the event of getting a 1,
Math 2p81
Lecture 3
Wai Kong (John) Yuen
[email protected]
Department of Mathematics, Brock University
September 11, 2013
Wai Kong (John) Yuen [email protected]
Math 2p81
Some counting
Thm 2.1. With m elements a1 , ., am and n elements
b1 , ., bn , it is poss
Math 2p81
Lecture 2
Wai Kong (John) Yuen
[email protected]
Department of Mathematics, Brock University
September 15, 2013
Wai Kong (John) Yuen [email protected]
Math 2p81
Set notation review
Sets A, B, C , . Elements a1 , a2 , a3 , .
Universal sets S. Empty s
Math 2p81
Lecture 1
Wai Kong (John) Yuen
[email protected]
Department of Mathematics, Brock University
September 4, 2013
Wai Kong (John) Yuen [email protected]
Math 2p81
What is probability?
A measure of ones belief in the occurrence of a future event.
The mo
Math 2p81
Lecture 15
Wai Kong (John) Yuen
[email protected]
Department of Mathematics, Brock University
November 1, 2013
Wai Kong (John) Yuen [email protected]
Math 2p81
Continuous rv
Discrete rvs are useful but many experiments have outcomes
with a continuou
Math 2p81
Lecture 13
Wai Kong (John) Yuen
[email protected]
Department of Mathematics, Brock University
October 22, 2013
Wai Kong (John) Yuen [email protected]
Math 2p81
Proof of Thm 3.2
Thm 3.2. Let Y be a rv and g : R R be a real-valued
function. Then U = g
Math 2p81
Lecture 14
Wai Kong (John) Yuen
[email protected]
Department of Mathematics, Brock University
October 29, 2013
Wai Kong (John) Yuen [email protected]
Math 2p81
Exercise 3.139
In the daily production of a certain kind of rope, the number
of defects p
Math 2p81
Lecture 17
Wai Kong (John) Yuen
[email protected]
Department of Mathematics, Brock University
November 13, 2013
Wai Kong (John) Yuen [email protected]
Math 2p81
Managing risks
In many applications, it is often of interest to nd a value c so
that P(Y