1. SGG2413 - Theory of Probability1 Probability Distribution 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 Spatial Statistics (SGG 2413)

2. SGG2413 - Theory of Probability2 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

3. SGG2413 - Theory of Probability3 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

4. SGG2413 - Theory of Probability4 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

5. SGG2413 - Theory of Probability5 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

6. SGG2413 - Theory of Probability6 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.)

7. SGG2413 - Theory of Probability7 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.)

8. SGG2413 - Theory of Probability8 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 SxSx ζ X(ζ) = x

9. SGG2413 - Theory of Probability9 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

10. SGG2413 - Theory of Probability10 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.  s  S p(s) = 1 Basic concepts of random variables (cont.)

11. SGG2413 - Theory of Probability11 Outcome, xProb. xCumulative prob. X 11/6 = 0.166 21/62/6= 0.333 31/63/6= 0.500 41/64/6= 0.666 51/65/6= 0.833 61/66/6= 1.000 E.g. rolling a dice… Basic concepts of random variables (cont.)

12. SGG2413 - Theory of Probability12 Types of Random Variables Continuous –Probability Density Function Discrete –Probability Mass Function Marginal change: Bounded area: No marginal change: No bounded area:

13. SGG2413 - Theory of Probability13 f X (x) dx f X (x) x Types of Random Variables - continuous

14. SGG2413 - Theory of Probability14 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 532

15. SGG2413 - Theory of Probability15 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 Types of plant coleoptera Plant APlant B 53 2

16. SGG2413 - Theory of Probability16 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)

17. SGG2413 - Theory of Probability17 Types of plant coleoptera Plant A Plant B 532 P(A|B) P(A and B) = P(A) P(B|A) = (5/10)(2/10) = 0.5 x 0.2 = 0.1 Types of plant coleoptera Plant APlant B 53 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:

18. SGG2413 - Theory of Probability18 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: Discrete probability distribution Xnxnx P(x) 2200.2 3400.4 4300.3 5100.1 Total1001.0

19. SGG2413 - Theory of Probability19 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

20. SGG2413 - Theory of Probability20 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.

21. SGG2413 - Theory of Probability21 TreeNo. of fruit landings (X i ) No. of fruits with borers attack (f Xi ) Prob. of fruit landings (p Xi = f Xi /  X i ) Expected no. fruits with borers (f Xi x p Xi )(f Xi –mean) 2 (f Xi –mean) 2 x p Xi 12080.131.05 1.420.18 21060.070.39 0.660.04 31540.100.39 7.900.77 41460.090.55 0.660.06 52480.161.25 1.420.22 620120.131.57 26.933.52 71890.121.06 4.790.56 81040.070.26 7.900.52 91420.090.18 23.142.12 10820.050.10 23.141.21 Sum153611.00 Mean 6.81 Variance 9.21 Std. dev. 3.04 Discrete probability distribution – Maduria magniplaga

22. SGG2413 - Theory of Probability22 Expected no. of fruits with borers: E(X i ) =  X.px =  (f Xi.X i /  X i ) = 6.81 ≈ 7 Variance of fruit borers’ attack: ● Standard deviation of fruit borers’ attack:  2 = E[(X-E(X)) 2 ] =  (f ni – mean) 2 x p Xi  =  9.21 = 9.21 = 3.04 Discrete probability distribution – Maduria magniplaga

23. SGG2413 - Theory of Probability23 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

24. SGG2413 - Theory of Probability24 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.)

25. SGG2413 - Theory of Probability25 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 Discrete probability distribution: Binomial (cont.)

26. SGG2413 - Theory of Probability26 Cumulative Distribution Function Defined as the probability of the event {X≤x} Properties x 2 1 F x (x) ¼ ½ ¾ 10 3 1 x

27. SGG2413 - Theory of Probability27 Probability Density Function f X (x) dx f X (x) x The pdf is computed from Properties For discrete r.v

28. SGG2413 - Theory of Probability28 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 A i mutally exclusive and exhaustive events

29. SGG2413 - Theory of Probability29 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

30. SGG2413 - Theory of Probability30 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

31. SGG2413 - Theory of Probability31 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

32. SGG2413 - Theory of Probability32 Thank you