Classnotes2 - 2 Random Experiments Sample Spaces • D fi i...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 2 Random Experiments Sample Spaces • D fi i i Definition: • D fi i i Definition: 3 • 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 More “Sample Space” Examples • Example 2-1 (continued) Example 2-1 5 Discrete v. Continuous Sample v Spaces 6 More “Sample Space” Examples Example 2-2 • 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 8 The Outcomes in a Sample Space Are Not Necessarily Equally Likely y q y y More “Sample Space” Examples • In general, the outcomes in a sample space are not equally likely: Example 2-2 (extended) – 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): • • Where “n” means a failed connector and “y” an acceptable one • In this case, the sample space is discrete ( the , p p (as number of outcomes is countably infinite) 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 9 Sampling With (and Without) Replacement Q. what’s the sample space if there is only a single defective item in the population? 10 Visualizing a Sample Space via a Tree Digram • In experiments where samples are taken from a population, it is significant whether or not this is done with or without replacement: • 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: – 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 – 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 • 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 12 A Tree Diagram Example A Tree Diagram Example Example 2-3 Figure 2-5 Tree diagram for three messages. 13 Events 14 Event (Set) Operators • 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 16 An Event Example Mutually Exclusive Events • D fi i i Definition: Example 2-6 • This says that the intersection of two mutually exclusive events is the empty set – I.e. these events have no outcomes in common 17 Visualizing Sample Spaces and Events Using Venn Diagrams 18 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: Figure 2-8 Venn diagrams. 19 20 More Counting Techniques: Permutations A Permutations Example Permutations: Permutations of subsets: Example 2-10 Permutations of subsets: 21 Further Permutations 22 Another Permutations Example Permutations of Similar Objects Permutations of Similar Objects: Example 2-11 23 24 More Counting Techniques: Combinations Another Permutations Example Permutations of Similar Objects: Example 2-12 Combinations 25 A Combinations Example 26 Interpretations of Probability • Two philosophical approaches are commonly taken: Combinations: Example 2-13 – Frequentist (aka objective): • The relative frequency of occurrence, in a long run , yp ( g p of trials, of some type of event (e.g. a coin turns up “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) 27 28 Dealing With Discrete Sample Spaces A Simple Frequentist Example • 2-2.1 Introduction • For the special case where each member of a sample space is equally likely to occur: • To determine the probability of an event: Figure 2-10 Relative frequency of corrupted pulses sent over a communications channel. 29 A Simple Example on Event Probability (Example 2 15) 2-15 30 Another Example on Event Probability Example 2-16 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 32 The Axioms of Mathematical Probability Addition Rules The probability of the union of two events: The probabilit of the union of three e ents: probability nion events: 33 A More General Definition of Mutually Exclusive Events 34 An Example of Four 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): Figure 2-12 Venn diagram of four mutually exclusive events 35 36 Visualizing This Example Using a Venn Diagram 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 Figure 2-13 Conditional probabilities for parts with surface flaws 37 Conditional Probability Computations 38 A Conditional Probability Example • The defining computation: Example 2-26 • The multiplication rule: 39 40 Using a Tree Diagram to Display Conditional Probabilities 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: • The data on 400 parts in the table below can be visualized in a tree diagram: i li d i di – 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 ( probability that the 1st is from m/c 1 and the 2nd from m/c 2? – 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 41 The Total Probability Rule 42 The Total Probability Rule(s) Total Probability Rule (for two events): • This gives a way of determining the probability of an event if enough conditional probabilities of the event are known Total Probability Rule (for multiple events): 43 44 A Total Probability Rule Example Independence Example 2-27: semiconductor failure (and contamination level): • The two-event case: • The multiple-event case: 45 A Simple Independence Example 46 Bayes’ Theorem • Preamble: rearrange eqn. 2-10 from earlier: • The theorem: 47 48 A Bayes’ Theorem Example (Ex. 2-37) Random Variables Definition: Distinguishing random variables from real experimental outcomes: 49 50 51 52 Types and Examples of Random Variable • Discrete v. continuous random variables: • Examples: l ...
View Full Document

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.

Ask a homework question - tutors are online