Transcript USC3002_2008.Lect3 - Department of Mathematics

USC3002 Picturing the World
Through Mathematics
Wayne Lawton
Department of Mathematics
S14-04-04, 65162749 [email protected]
Theme for Semester I, 2008/09 : The Logic of
Evolution, Mathematical Models of Adaptation
from Darwin to Dawkins
MOTIVATION
Probability and Statistics play an increasingly
crucial role in evolution research
http://www.springer.com/east/home/life+sci/bioinformatics?
SGWID=5-10031-22-34952257-0
http://www-stat.stanford.edu/~susan/courses/s366/
http://findarticles.com/p/articles/mi_qa3746/is_199
904/ai_n8829021/pg_16
REFERENCES
[1] Rudolph Carnap, An Introduction
to the Philosophy of Science, Dover, N.Y., 1995.
[2] Leong Yu Kang, Living With Mathematics,
McGraw Hill, Singapore, 2004. (GEM Textbook)
(1 Reasoning, 2 Counting, 3 Graphing, 4 Clocking,
5 Coding, 6 Enciphering, 7 Chancing, 8 Visualizing)
MATLAB Demo Random Variables & Distributions
Discuss Topics in Chap. 2-4 in [1], Chap. 1, 7 in [2].
Baye’s Theorem & The Envelope Problem,
Deductive, Inductive, and Abductive Reasoning.
Assign computational tutorial problems.
RANDOM VARIABLES
The number that faces up on an ‘unloaded’ dice rolled
on a flat surface is in the set { 1, 2, 3, 4, 5, 6 } and the
probability of each number is equal and hence = 1/6
After rolling a dice, the number is fixed to those who
know it but remains an unknown, or random variable
to those who do not know it. Even while it is still
rolling, a person with a laser sensor connected with a
sufficiently powerful computer may be able to predict
with some accuracy the number that will come up.
This happened and the Casino was not amused !
MATLAB PSEUDORANDOM VARIABLES
The MATLAB (software) function rand generates
decimal numbers d / 10000 that behaves as if d is a
random variable with values in the set {0,1,2,…,9999}
with equal probability. It is a pseudorandom variable.
It provides an approximation of a random variable x
with values in the interval [0,1] of real numbers such
that for all 0 < a < b < 1 the probability that x is in the
interval [a,b] equals b-a = length of [a,b]. These are
called uniformly distributed random variables.
PROBABILITY DISTRIBUTIONS
Random variables with values in a set of integers
are described by discrete distributions
Uniform (Dice), Prob(x = k) = 1/6 for k = 1,…,6
Binomial Prob(x = k) = a^k (1-a)^(n-k) n!/(n-k)!k!
for k = 0,1,…,n where an event that has probability
a occurs k times out of a maximum of n times and
k! = 1*2…*(k-1)*k is called k factorial.
Poisson Prob(x = k) = a^k exp(-a) / k! for k > -1
where k is the event that k-atoms of radium decay if
a is the average number of atoms expected to decay.
PROBABILITY DISTRIBUTIONS
Random variables with values in a set of real
numbers are described by continuous distributions
Uniform over the interval [0,1]
b
Prob( x [a, b])   1dx  b  a for 0  a  b  1
a
Gaussian or Normal
b


Prob( x [a, b])  
exp 2 2 dx
here   mean
2
and   standard deviation,   variance
1
a  2
( x   ) 2
MATLAB HELP COMMAND
>> help rand
RAND Uniformly distributed random numbers.
RAND(N) is an N-by-N matrix with random entries, chosen
from a uniform distribution on the interval (0.0,1.0).
RAND(M,N) is a M-by-N matrix with random entries.
>> help hist
HIST Histogram.
N = HIST(Y) bins the elements of Y into 10 equally spaced
containers and returns the number of elements in each
container. If Y is a matrix, HIST works down the
columns.
N = HIST(Y,M), where M is a scalar, uses M bins.
MATLAB DEMONSTRATION 1
14
16
14
12
12
10
10
8
8
6
6
4
4
2
0
2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0
0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
16
15
14
12
10
10
8
6
5
4
2
0
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.2
0.3
0.4
0.5
0.6
0.7
Why do these histograms look different ?
0.8
0.9
1
MATLAB DEMONSTRATION 2
>> x = rand(10000,1);
>> hist(x,41)
300
250
200
150
100
50
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
MORE MATLAB HELP COMMANDS
>> help randn
RANDN Normally distributed random numbers.
RANDN(N) is an N-by-N matrix with random entries,
chosen from a normal distribution with mean zero,
variance one and standard deviation one.
RANDN(M,N) is a M-by-N matrix with random entries.
>> help sum
SUM Sum of elements.
For vectors, SUM(X) is the sum of the elements of X.
For matrices, SUM(X) is a row vector with the sum over
each column.
3 1
sum 
 7 6

4 5
MATLAB DEMONSTRATION 3
>> s = -4:.001:4;
>> plot(s,exp(s.^2/2)/(sqrt(2*pi)))
>> grid
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
-4
-3
-2
-1
0
1
2
3
4
MATLAB DEMONSTRATION 3
>> x = randn(10000,1);
>> hist(x,41)
800
700
600
500
400
300
200
100
0
-5
-4
-3
-2
-1
0
1
2
3
4
MATLAB DEMONSTRATION 3
>> x = rand(5000,10000);
>> y = sum(x);
>> hist(y,41)
800
700
600
500
400
300
200
100
0
2420
2440
2460
2480
2500
2520
2540
2560
2580
CENTRAL LIMIT THEOREM
The sum of N real-valued random variables
y = x(1) + x(2) + … + x(N) will be a random
variable. If the x(j) are independent and have the
same distribution then as N increases the
distributions of y will approach (means gets
closer and closer to) a Gaussian distribution.
The mean of this Gaussian distribution
= N times the (common) mean of the x(j)
The variance of this Gaussian distribution
= N times the (common) variance of the x(j)
CONDITIONAL PROBABILITY
Recall that on my dice the ‘numbers’ 1 and 4
are red and the numbers 2, 3, 5, 6 are blue.
I roll one dice without letting you see how it rolls.
What is the probability that I rolled a 4 ?
I repeat the procedure BUT tell you that the number
is red. What is the probability that I rolled a 4 ?
This probability is called the conditional probability
that x = 4 given that x is red (i.e. x in {1,4})
Prob( A | B)  Prob of event A given event B
CONDITIONAL PROBABILITY
If A and B are two events then A  B denotes the
event that BOTH event A and event B happen.
Common sense implies the following LAW:
Prob ( A  B)  Prob ( B)  Prob ( A | B)
Example Consider the roll of a dice. Let A be the
event x = 4 and let B be the event x is red (= 1 or 4)
Prob( A  B)  Pr ob( A)  1 / 6
Pr ob( B)  1 / 3, Prob( A | B)  1 / 2
Question What does the LAW say here ?
BAYE’s THEOREM
http://en.wikipedia.org/wiki/Bayes'_theorem
for an event A, A
c
denotes the event not A
Question Why does Prob(A)  Prob(A )  1 ?
c
Prob(A) and Prob(B) are called marginal distributions.
Question Why does
Prob( B)  Prob(B | A)  Prob(A)  Prob(B | A c )  Prob(A c )
Question Why does
Prob(B | A)  Prob(A)
Prob( B) 
Prob(B)
INDUCTIVE & ABDUCTIVE REASONING
http://en.wikipedia.org/wiki/Inductive_reasoning
Inductive reasoning is the process of reasoning in which the premises of an
argument support the conclusion but do not ensure it.
This is in contrast to Deductive reasoning in which the conclusion is necessitated
by, or reached from, previously known facts.
http://en.wikipedia.org/wiki/Abductive_reasoning
Abductive reasoning, is the process of reasoning to the best explanations.
In other words, it is the reasoning process that starts from a set of facts and
derives their most likely explanations.
The philosopher Charles Peirce introduced abduction into modern logic.
In his works before 1900, he mostly uses the term to mean the use of a known rule to
explain an observation, e.g., “if it rains the grass is wet” is a known rule used to explain
that the grass is wet. He later used the term to mean creating new rules to explain new
observations, emphasizing that abduction is the only logical process that actually
creates anything new. Namely, he described the process of science as a combination
of abduction, deduction and implication, stressing that new knowledge is only created
by abduction.
Homework 4. Due Monday 15.09.08
http://www.holah.karoo.net/experimental_method.htm
Carnap p. 41 [1] “One of the great distinguishing
features of modern science, as compared to the science
of earlier periods, is its emphasis on what is called the
“experimental method”. “
Question 1. Discuss how the experimental method
differs from the method of observation.
Question 2. Discuss the fields of inquiry that favor the
experimental methods and what fields do not and why.
Question 3. Describe the Ideal Gas Law and the
experimental to test it.
Homework 4. Due Monday 15.09.08
Question 4. The uniform distribution on [0,1] has mean ½
and variance 1/12. Use the Central Limit Theorem to compute
the mean and variance of the random variable y whose
histogram is shown in vufoil # 13.
Question 5. I roll a dice to get a random variable x in
{1,2,3,4,5,6}, then put x dollars in one envelope and put 2x in
another envelope then flip a coin to decide which envelope to
give you (so that you receive the smaller or larger amount with
equal probability). Use Baye’s Theorem to compute the
probability that you received the smaller amount
CONDITIONED on YOUR FINDING THAT YOU
HAVE 1,2,3,4,5,6,8,10,12 dollars. Then use these
conditional probabilities to explain the Envelope Paradox.