Classnotes_4 - 2 Random Experiments • D fi i i Definition 3 Sample Spaces • D fi i i Definition • E.g if we are to flip a(typical coin once

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Unformatted text preview: 2 Random Experiments • D fi i i Definition: 3 Sample Spaces • D fi i i Definition: • E.g. if we are to flip a (typical) coin once, the possible outcomes are “heads” or “tails” (that we heads tails can denote H and T respectively) to give: S={H,T} • If our experiment involves flipping a single coin twice t i we get: t S = { HH, HT, TH, TT } 4 More “Sample Space” Examples Example 2-1 5 More “Sample Space” Examples • Example 2-1 (continued) 6 Discrete v. Continuous Sample v Spaces • Discrete: – Th number of members of th sample space is fi it or The b f b f the l i finite countably infinite • C ti Continuous: – The number of members of the sample space is infinite and uncountable d t bl • Notes: – A “countably infinite” set has the same number of members as the set of positive integers (i.e. you can count them but it’ll take you forever) – An uncountable set has the same number of members as a portion of the real line (e.g. there’s an uncountable number of real numbers between 0 and 1) 7 More “Sample Space” Examples Example 2-2 8 More “Sample Space” Examples Example 2-2 (extended) • • Where “n” means a failed connector and “y” an acceptable one • In this case, the sample space is discrete (as the , p p ( number of outcomes is countably infinite) 9 The Outcomes in a Sample Space Are Not Necessarily Equally Likely y q y y • In general, the outcomes in a sample space are not equally likely: – If the coin used in our coin-flipping examples is “fair”, each outcome is pp g p f equally likely – In all of our other examples so far, the outcomes are not equally likely • Another example with unequally likely outcomes: – Suppose we have a population of 100 items, n of which are defective, and we draw a sample of two items – D Denoting a “good” item by “g” and a defective item by “d”, the sample i “ d” i b “ ” d d f i i b “d” h l space is as follows (provided that n>=2): S = {gg, gd, dg, dd} ... if the ordering of the samples is important or S = {gg, gd, dd} ... if the ordering of the samples is unimportant Q. what’s the sample space if there is only a single defective item in the population? 10 Sampling With (and Without) Replacement • In experiments where samples are taken from a population, it is significant whether or not this is done with or without replacement: – When sampling with replacement, the sampled item is replaced into the population before the next sample is drawn – When sampling without replacement, sampled items are not returned to the population t d t th l ti • E.g. if two items are sampled ( g p (one at a time) from the ) population {a, b, c} the sample space would be one of: – Swithout = {ab, ac, ba, bc, ca, cb} – Swith = {aa, ab, ac, bb, ba, bc, cc, ca, cb} 11 Visualizing a Sample Space via a Tree Digram • When a sample space can be constructed in several steps (or stages) we can represent it using a tree diagram stages), • Construction procedure for a tree diagram: – Start at the root of the tree (though this is typically at the top of the diagram, not the bottom) – Draw a branch (from the root) to represent each of the n1 outcomes at the first stage – From the end of each branch, draw a branch to represent each of the n2 outcomes at the second stage – Continue in a similar fashion for all subsequence stages 12 A Tree Diagram Example Example 2-3 13 A Tree Diagram Example Figure 2-5 Tree diagram for three messages. 14 Events • D fi i i Definition: • • An event typically represents a collection of related outcomes that may be of signifcance • Since an event is a set (remember the sample (remember, space is a set), we can manipulate events using set operators to form other events of significance 15 Event (Set) Operators 16 An Event Example Example 2-6 17 Mutually Exclusive Events • D fi i i Definition: • This says that the intersection of two mutually exclusive events is the empty set – I.e. these events have no outcomes in common 18 Visualizing Sample Spaces and Events Using Venn Diagrams Figure 2-8 Venn diagrams. 19 Determining the Size of a Sample Space • The term counting techniques is used to refer to formulae for computing the size of a sample space (or event) • One such technique is the multiplication rule: 20 More Counting Techniques: Permutations Permutations: Permutations of subsets: 21 A Permutations Example Permutations of subsets: Example 2-10 22 Further Permutations Permutations of Similar Objects 23 Another Permutations Example Permutations of Similar Objects: Example 2-11 24 Another Permutations Example Permutations of Similar Objects: Example 2-12 25 More Counting Techniques: Combinations Combinations 26 A Combinations Example Combinations: Example 2-13 27 Interpretations of Probability • Two philosophical approaches are commonly taken: – Frequentist (aka objective): • The relative frequency of occurrence, in a long run of trials, of some type of event (e.g. a coin turns up , yp ( g p “heads”) – Subjective (aka Bayesian): j ( y ) • The degree of belief in a statement, or the extent to which it is supported by the available evidence (e.g. the Calgary Flames will win the 2009 home opener) 28 A Simple Frequentist Example • 2-2.1 Introduction Figure 2-10 Relative frequency of corrupted pulses sent over a communications channel. 29 Dealing With Discrete Sample Spaces • For the special case where each member of a sample space is equally likely to occur: • To determine the probability of an event: 30 A Simple Example on Event Probability (Example 2 15) 2-15 Let th L t the event E be that one of the 30 diodes meeting power t b th t f th di d ti requirements is chosen, then P(E) is determined as follows: 31 Another Example on Event Probability Example 2-16 32 The Axioms of Mathematical Probability 33 Addition Rules The probability of the union of two events: The probabilit of the union of three e ents: probability nion events: 34 A More General Definition of Mutually Exclusive Events • Slide #18 defined mutual exclusivity for two events • The concept generalizes to the case of k events as follows (where 1 ” i ” k; 1 ” j”k; ij): 35 An Example of Four Mutually Exclusive Events Figure 2-12 Venn diagram of four mutually exclusive events 36 Conditional Probability • This concept deals with how the probability of some event A should be reevaluated if we know that some other event B has occurred • Consider a manufacturing process example: – 10% of items produced contain a visible surface flaw and 25% of these are functionally defective – 5% of the parts without a visible surface flaw are functionally defective – Let D denote the event that a part is defective and let F denote the event that a part has a surface flaw – Then we let P(D|F) denote the conditional probability of D given Then, F, i.e. the probability that a part is defective, given that the part has a surface flaw 37 Visualizing This Example Using a Venn Diagram Figure 2-13 Conditional probabilities for parts with surface flaws 38 Conditional Probability Computations • The defining computation: • The multiplication rule: 39 A Conditional Probability Example Example 2-26 40 Using a Tree Diagram to Display Conditional Probabilities • The data on 400 parts in the table below can be visualized in a tree diagram: i li d i di 41 Random Samples and Conditional Probability • To select from a batch randomly implies that at each step of the sample the items that remain in the batch are sample, equally likely to be selected • Example: – 50 parts in total, 10 from machine 1 and 40 from machine 2 p y (without replacement) what’s the p ) – If 2 parts are selected randomly ( st is from m/c 1 and the 2nd from m/c 2? probability that the 1 – Let E1=first part selected is from machine 1; E2=2nd part selected is from machine 2 – Since the sampling is random, P(E1)=10/50 and P(E2|E1)=40/49 – Thus P(E1 ŀ E2) = P(E2|E1) . P(E1) = 40/49 . 10/50 = 8/49 42 The Total Probability Rule • This gives a way of determining the probability of an event if enough conditional probabilities of the event are known 43 The Total Probability Rule(s) Total Probability Rule (for two events): Total Probability Rule (for multiple events): 44 A Total Probability Rule Example Example 2-27: semiconductor failure (and contamination level): 45 Independence • The two-event case: • The multiple-event case: 46 A Simple Independence Example 47 Bayes’ Theorem • Preamble: rearrange eqn. 2-10 from earlier: • The theorem: 48 A Bayes’ Theorem Example (Ex. 2-37) 49 Random Variables Definition: Distinguishing random variables from real experimental outcomes: 50 Types and Examples of Random Variable • Discrete v. continuous random variables: • Examples: l 51 52 ...
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This note was uploaded on 05/31/2010 for the course ENGG 319 taught by Professor Nanayakkara during the Fall '08 term at University of Calgary.

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