Probability Distribution

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

2. 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 SGG2413 - Theory of Probability

3. 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 SGG2413 - Theory of Probability

4. 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 SGG2413 - Theory of Probability

5. 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 SGG2413 - Theory of Probability

6. 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 SGG2413 - Theory of Probability

7. 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 SGG2413 - Theory of Probability

8. S ζ X(ζ) = x x Sx 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. SGG2413 - Theory of Probability

9. 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 SGG2413 - Theory of Probability

10. Basic concepts of random variables (cont.) • 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 SGG2413 - Theory of Probability

11. Basic concepts of random variables (cont.) E.g. rolling a dice… SGG2413 - Theory of Probability

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

13. fX(x) dx x fX(x) Types of Random Variables - continuous SGG2413 - Theory of Probability

14. Plant A Plant B 5 2 3 Types of plant coleoptera 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 SGG2413 - Theory of Probability

15. Plant A Plant B 5 3 2 Types of plant coleoptera 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 SGG2413 - Theory of Probability

16. 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) SGG2413 - Theory of Probability

17. Plant A Plant B 5 3 2 Types of plant coleoptera P(A|B) Plant A Plant B 5 2 3 Types of plant coleoptera A & B Statistically dependent: A & B Statistically independent: P(A and B) = P(A) P(B|A) = (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 SGG2413 - Theory of Probability

18. Discrete probability distribution • 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: SGG2413 - Theory of Probability

19. The expected value or mean of X is Properties The variance of X is The standard deviation of X is Properties Expected Value and Variance continuous discrete SGG2413 - Theory of Probability

20. 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. More on Mean and Variance SGG2413 - Theory of Probability

21. Discrete probability distribution – Maduria magniplaga SGG2413 - Theory of Probability

22. 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 SGG2413 - Theory of Probability

23. 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). SGG2413 - Theory of Probability

24. where Mean and variance: 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 SGG2413 - Theory of Probability

25. 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 SGG2413 - Theory of Probability

26. Cumulative Distribution Function • Defined as the probability of the event {X≤x} • Properties Fx(x) 1 x Fx(x) 1 ¾ ½ ¼ 3 x 0 1 2 SGG2413 - Theory of Probability

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

28. 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 Conditional Distribution SGG2413 - Theory of Probability

29. 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 Joint Distributions SGG2413 - Theory of Probability

30. 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 Conditional Probability and Expectation SGG2413 - Theory of Probability

31. 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 Independence of two Random Variables SGG2413 - Theory of Probability

32. Thank you SGG2413 - Theory of Probability