SP2170 Doing Science Lecture 3: Random Variables, Distributions, Inductive &amp; Abductive Reasoning, Experiments

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SP2170 Doing Science Lecture 3: Random Variables, Distributions, Inductive &amp; Abductive Reasoning, Experiments. Wayne M. Lawton Department of Mathematics National University of Singapore 2 Science Drive 2 Singapore 117543. Email [email protected]

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### SP2170 Doing ScienceLecture 3: Random Variables, Distributions, Inductive & Abductive Reasoning, Experiments

Wayne M. Lawton

Department of Mathematics

National University of Singapore

2 Science Drive 2

Singapore 117543

Email [email protected]

Tel (65) 6516-2749

1

PLAN FOR WEEK 3 LECTURE

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.

2

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 !

3

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.

4

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.

5

PROBABILITY DISTRIBUTIONS

Random variables with values in a set of real

numbers are described by continuous distributions

Uniform

Gaussian or Normal

here

and

6

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.

7

MATLAB DEMONSTRATION 1

8

Why do these histograms look different ?

MATLAB DEMONSTRATION 2

>> x = rand(10000,1);>> hist(x,41)

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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.

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MATLAB DEMONSTRATION 3

>> s = -4:.001:4;>> plot(s,exp(s.^2/2)/(sqrt(2*pi)))>> grid

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MATLAB DEMONSTRATION 3

>> x = randn(10000,1);>> hist(x,41)

12

MATLAB DEMONSTRATION 3

>> x = rand(5000,10000);>> y = sum(x);>> hist(y,41)

13

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)

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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})

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CONDITIONAL PROBABILITY

If A and B are two events then

denotes the

event that BOTH event A and event B happen.

Common sense implies the following LAW:

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)

Question What does the LAW say here ?

16

BAYE’s THEOREM

http://en.wikipedia.org/wiki/Bayes\'_theorem

for an event A,

denotes the event not A

Question Why does

Prob(A) and Prob(B) are called marginal distributions.

Question Why does

Question Why does

17

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.

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EXPERIMENTS

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 How does the experimental method differ

from the method of observation ?

Question What fields favor the experimental methods

and what fields do not and why ?

Ideal Gas Law - one of the greatest experiments !

/Lecture_3.ppt Slides 1-5

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TUTORIAL QUESTIONS

Question 1. 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 2. 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.

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