Probability Distribution

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Probability Distribution
Spatial Statistics (SGG 2413)

• Assoc. Prof. Dr. Abdul Hamid b. Hj. Mar Iman
• Former Director
• Centre for Real Estate Studies
• Faculty of Engineering and Geoinformation Science
• Universiti Tekbnologi Malaysia
• Skudai, Johor

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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

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Contents

• 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

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Basic probability theory

• 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

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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

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Basic probability theory (cont.)

• 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

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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

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Random Variables

• Definition
• A function of changeable and measurable
  characteristic, X, which assigns a real number
  X(?) to each outcome ? 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.

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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

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Basic concepts of random variables (cont.)

• E.g. rolling a dice
• SpaceS 1, 2, 3, 4, 5, 6
• EventOdd numbers A 1, 3, 5
• Even numbers B 2,4,6
• Sample point1, 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.
• ?s?S p(s) 1

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Basic concepts of random variables (cont.)
E.g. rolling a dice
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Types of Random Variables

• Discrete
• Probability Mass Function

• Continuous
• Probability Density Function

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Types of Random Variables - continuous
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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
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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
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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(BA)
• where P(BA) is the probability of B
  conditioned on A.
• If A and B are statistically independent
• P(BA) P(B) and then
• P(A and B) P(A) P(B)

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P(AB)
Plant A
Plant B
5
3
2
Types of plant coleoptera
A B Statistically dependent
A B Statistically independent
P(A and B) P(A) P(BA)
(5/10)(2/10) 0.5 x 0.2
0.1
P(A and B) P(A) P(B)
(5/10)(3/10) 0.5 x 0.3
0.15
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Discrete probability distribution

• Lets define x no. of bedroom of sampled houses
• Lets x 2, 3, 4, 5
• Also, lets probability of each outcome be

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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

continuous
discrete
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More on Mean and Variance

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

• Physical Meaning
• If pmf is a set of point masses, then the
  expected value µ is the center of mass, and the
  standard deviation s is a measure of how far
  values of x are likely to depart from µ

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Discrete probability distribution Maduria
magniplaga
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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

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Discrete probability distribution Binomial

• 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).

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Discrete probability distribution Binomial
(cont.)
The probability that a random variable X B(n,
p) is equal to the value k, where k 0, 1,, n
is given by
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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 14
• ?2 np(1-p) 14 x (1-0.78) 3.08

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Cumulative Distribution Function

• Defined as the probability of the event Xx
• Properties

Fx(x)
1
x
Fx(x)
1
¾
½
¼
2
1
0
3
x
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Probability Density Function
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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

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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

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Conditional Probability and Expectation

• The conditional CDF of Y given the event Xx is
• The conditional PDF of Y given the event Xx is

• The conditional expectation of Y given Xx is

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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

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Thank you