SGG2413 - Theory of Probability1 Probability Distribution Assoc. Prof. Dr. Abdul Hamid b. Hj. Mar...

SGG2413 - Theory of Probability 1

Probability Distribution

Assoc. Prof. Dr. Abdul Hamid b. Hj. Mar Iman

Former DirectorCentre for Real Estate Studies

Faculty of Engineering and Geoinformation ScienceUniversiti Tekbnologi Malaysia

Skudai, Johor

Spatial Statistics (SGG 2413)


Learning Objectives

• Overall: To expose students to the concepts of probability

• Specific: Students will be able to: * define what are probability and random variables * explain types of probability * write the operational rules in probability * understand and explain the concepts of probability distribution


• Basic probability theory• Random variables• Addition and multiplication rules of probability• Discrete probability distribution: Binomial

probability distribution, Poisson probability distribution

• Continuous probability distribution• Normal distribution and standard normal

distribution• Joint probability distribution

Contents


• Probability theory examines the properties of random variables, using the ideas of random variables, probability & probability distributions.

• Statistical measurement theory (and practice) uses probability theory to answer concrete questions about accuracy limits, whether two samples belong to the same population, etc.

probability theory is central to statistical analyses

Basic probability theory


Basic probability theory

• Vital for understanding and predicting spatial patterns, spatial processes and relationships between spatial patterns

• Essential in inferential statistics: tests of hypotheses are based on probabilities

• Essential in the deterministic and probabilistic processes in geography: describe real world processes that produce physical or cultural patterns on our landscape


• Deterministic process – an outcome that can be predicted almost with 100% certainty.

• E.g. some physical processes: speed of comet fall, travel time of a tornado, shuttle speed

• Probabilistic process – an outcome that cannot be predicted with a 100% certainty

• Most geographic situations fall into this category due to their complex nature

• E.g. floods, draught, tsunami, hurricane• Both categories of process is based on random

variable concept

Basic probability theory (cont.)


• Random probabilistic process – all outcomes of a process have equal chance of occurring. E.g.

* Drawing a card from a deck, rolling a die, tossing

a coin

…maximum uncertainty

• Stochastic processes – the likelihood of a particular outcome can be estimated. From totally random to totally deterministic. E.g.

* Probability of floods hitting Johor: December vs. January

…probability is estimated based on knowledge which will

affect the outcome

Basic probability theory (cont.)


Random Variables

• Definition:– A function of changeable and

measurable characteristic, X, which assigns a real number X(ζ) to each outcome ζ in the sample space of a random experiment

• Types of random variables:– Continuous. E.g. income, age,

speed, distance, etc.– Discrete. E.g. race, sex,

religion, etc.

S

x

Sx

ζ

X(ζ) = x


Basic concepts of random variables

• Sample Point– The outcome of a random experiment

• Sample Space, S– The set of all possible outcomes– Discrete or continuous

• Events– A set of outcomes, thus a subset of S– Certain, Impossible and Elementary


• E.g. rolling a dice… Space…S = {1, 2, 3, 4, 5, 6} Event…Odd numbers: A = {1, 3, 5} …Even numbers: B = {2,4,6} Sample point…1, 2,..• Let S be a sample space of an experiment with a

finite or countable number of outcomes. • We assign p(s) to each outcome s.• We require that two conditions be met: 0 p 1 for each s S.

sS p(s) = 1

Basic concepts of random variables (cont.)


Outcome, x Prob. x Cumulative prob. X

1 1/6 1/6 = 0.166

2 1/6 2/6 = 0.333

3 1/6 3/6 = 0.500

4 1/6 4/6 = 0.666

5 1/6 5/6 = 0.833

6 1/6 6/6 = 1.000

E.g. rolling a dice…

Basic concepts of random variables (cont.)


Types of Random Variables

• Continuous– Probability Density

Function

• Discrete– Probability Mass

Function

X k kP x P X x

X X k kk

F x P x u x x

XX

dF xf x

dx

x

X XF x f t dt

Marginal change:

Bounded area:

No marginal change:

No bounded area:


fX(x)

dx

fX(x)

XP x X x dx f x dx

x

Types of Random Variables - continuous


Probability: Law of Addition If A and B are not mutually exclusive events: P(A or B) = P(A) + P(B) – P(A and B)E.g. What is the probability of types of coleoptera found on plant A or plant B?

P(A or B) = P(A) + P(B) – P(A and B)

= 5/10 + 3/10 – 2/10

= 6/10

= 0.6

Types of plant coleoptera

Plant A Plant B

5 32


Probability: Law of Addition (cont.)

If A and B are mutually exclusive events: P(A or B) = P(A) + P(B) E.g. What is the probability of types of coleoptera found on plant A or plant B?

P(A or B) = P(A) + P(B)

= 5/10 + 3/10

= 8/10

= 0.8


Plant A Plant B

5 3

2


Probability: Law of Multiplication If A and B are statistically dependent, the probability that A and B occur together: P(A and B) = P(A) P(B|A)

where P(B|A) is the probability of B conditioned on A.

If A and B are statistically independent:

P(B|A) = P(B) and then

P(A and B) = P(A) P(B)



Plant A Plant B

5 32

P(A|B)

P(A and B) = P(A) P(B|A) = (5/10)(2/10) = 0.5 x 0.2 = 0.1


Plant A Plant B

5 3

2

P(A and B) = P(A) P(B) = (5/10)(3/10) = 0.5 x 0.3 = 0.15

A & B Statistically dependent: A & B Statistically independent:


• Let’s define x = no. of bedroom of sampled houses• Let’s x = {2, 3, 4, 5} • Also, let’s probability of each outcome be:

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

2 3 4 5

No. of bedroom, x

Pro

ba

bili

ty, p

(x)

Discrete probability distribution

X nx P(x)

2 20 0.2

3 40 0.4

4 30 0.3

5 10 0.1

Total 100 1.0


Expected Value and Variance

• The expected value or mean of X is

• Properties

• The variance of X is

• The standard deviation of X is

• Properties

XE X tf t dt

k X kk

E X x P x

E c c

E cX cE X

E X c E X c

22Var X E X E X

Std X Var X

0Var c

2Var cX c Var X

Var X c Var X

continuous

discrete


More on Mean and Variance

• Physical Meaning– If pmf is a set of point

masses, then the expected value μ is the center of mass, and the standard deviation σ is a measure of how far values of x are likely to depart from μ

• Markov’s Inequality

• Chebyshev’s Inequality

• Both provide crude upper bounds for certain r.v.’s but might be useful when little is known for the r.v.

E XP X a

a

2

2P X a

a

2

1P X k

k


Tree No. of fruit landings

(Xi)

No. of fruits with

borers attack

(fXi)

Prob. of fruitlandings

(pXi = fXi/Xi)

Expected no. fruits with

borers

(fXi x pXi) (fXi –mean)2

(fXi –mean)2 x pXi

1 20 8 0.13 1.05 1.42 0.18

2 10 6 0.07 0.39 0.66 0.04

3 15 4 0.10 0.39 7.90 0.77

4 14 6 0.09 0.55 0.66 0.06

5 24 8 0.16 1.25 1.42 0.22

6 20 12 0.13 1.57 26.93 3.52

7 18 9 0.12 1.06 4.79 0.56

8 10 4 0.07 0.26 7.90 0.52

9 14 2 0.09 0.18 23.14 2.12

10 8 2 0.05 0.10 23.14 1.21

Sum 153 61 1.00

Mean 6.81

Variance 9.21

Std. dev. 3.04

Discrete probability distribution – Maduria magniplaga


• Expected no. of fruits with borers:

E(Xi) = X.px

= (fXi.Xi/Xi) = 6.81 ≈ 7• Variance of fruit borers’ attack: ● Standard deviation of fruit borers’ attack: 2 = E[(X-E(X))2]

= (fni – mean)2 x pXi = 9.21 = 9.21 = 3.04

Discrete probability distribution – Maduria magniplaga


• Outcomes come from fixed n random occurrences, X

• Occurrences are independent of each other

• Has only two outcomes, e.g. ‘success’ or

• ‘failure’

• The probability of "success" p is the same for each occurrence

• X has a binomial distribution with parameters n and p, abbreviated X ~ B(n, p).

Discrete probability distribution: Binomial


where

Mean and variance:

The probability that a random variable X ~ B(n, p) is equal to the value k, where k = 0, 1,…, n is given by

Discrete probability distribution: Binomial (cont.)


• E.g. The Road Safety Department discovered that the number of potential accidents at a road stretch was 18, of which 4 are fatal accidents. Calculate the mean and variance of the non-fatal accidents.

= np = 18 x 0.78 = 142 = np(1-p) = 14 x (1-0.78) = 3.08

Discrete probability distribution: Binomial (cont.)


Cumulative Distribution Function• Defined as the probability of the event {X≤x}

• Properties

XF x P X x

0 1XF x

lim 1XxF x

lim 0Xx

F x

if then X Xa b F a F a

X XP a X b F b F a

1 XP X x F x

x

2

1

Fx(x)

¼

½

¾

10 3

1

Fx(x)

x


Probability Density Function

fX(x)

dx

fX(x)

XP x X x dx f x dx

x

XX

dF xf x

dx

b

XaP a X b f x dx

x

X XF x f t dt

1 Xf t dt

X X k kk

f x P x x x

• The pdf is computed from

• Properties

• For discrete r.v


Conditional Distribution

• The conditional distribution function of X given the event B

• The conditional pdf is

• The distribution function can be written as a weighted sum of conditional distribution functions

where Ai mutally exclusive and exhaustive events

|X

P X x BF x B

P B

|| X

X

dF x Bf x B

dx

1

| |n

X X i ii

F x B F x A P A


Joint Distributions

• Joint Probability Mass Function of X, Y

• Probability of event A

• Marginal PMFs (events involving each rv in isolation)

• Joint CMF of X, Y

• Marginal CMFs ,

,

XY j k j j

j k

p x y P X x Y y

P X x Y y

, ,XY XY j kj A k A

P X Y A p x y

1

,XY j j XY j kk

p x P X x p x y

1 1 1 1, ,XYF x y P X x Y y

,X XYF x F x P X x

,Y XYF y F y P Y y


Conditional Probability and Expectation

• The conditional CDF of Y given the event {X=x} is

• The conditional PDF of Y given the event {X=x} is

• The conditional expectation of Y given X=x is

, ' '

|

y

XY

YX

f x y dyF y x

f x

,| XY

YX

f x yf Y x

f x

|| X Y

YX

f x y f yf y x

f x

| |YE Y x yf y x dy


Independence of two Random Variables

• X and Y are independent if {X ≤ x} and {Y ≤ y} are independent for every combination of x, y

• Conditional Probability of independent R.V.s

,XY X YF x y F x F y

,XY X Yf x y f x f y

,XY X Yf x y f x f y

|Y Yf y x f y

|X Xf x y f x


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SGG2413 - Theory of Probability1 Probability Distribution Assoc. Prof. Dr. Abdul Hamid b. Hj. Mar...

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