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

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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 ...
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