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Statistical processes 1 Probability Introduction.

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1 statistical processes 1 Probability Introduction

2 statistical processes 2 Class 2 Readings & Problems Reading assignment M & S Chapter 3 - Sections (Probability) Recommended Problems M & S Chapter 3 1, 20, 25, 29, 33, 57, 75, and 83

3 statistical processes 3 Introduction to Probability Probability - a useful tool Inferential statistics Infer population parameters probabilistically Stochastic modeling (engineering applications) Decision analysis Simulation Reliability Statistical process control Others …

4 statistical processes 4 Development of Probability Theory Chapter 3 - Introduction to probability Basic concepts Chapter 4 - Discrete random variables What is a random variable??? What is a discrete random variable??? Chapter 5 - Continuous random variable What is a continuous random variable??? Do not be afraid of random variables!!

5 statistical processes 5 What Is Probability? Deterministic models All parameters known with certainty Stochastic models One or more parameters are uncertain May be unknown Known but may take on more than 1 value Measure of uncertainty  probability Probability quantifies uncertainty!

6 statistical processes 6 Probability Most Common Viewpoint Frequentist view Probability is relative frequency of occurrence Most often associated with probability Adopted in textbook Probability inherent to physical process Property of large number (  ) of trials Examples of applications??

7 statistical processes 7 Probability An Alternative Perspective Bayesian view (aka personalist or subjective) Many real world applications not amenable to frequentist viewpoint What is probability of permanent lunar colony by 2015? What if asked in 1970? What if asked in 1998? What if asked in 2004??! Is probability here a property inherent to physical process?

8 statistical processes 8 Bayesian Probability What is key? What is probability RPI beat Cornell in hockey February 1971? RPI was ECAC champ that year What is probability RPI beat Cornell in hockey February 1971? The score was RPI 3, Cornell 1 State of knowledge defines probability

9 statistical processes 9 Frequentist Probability Building a Foundation Experiment Process of obtaining observations What are examples? Basic outcome A simple event Elemental outcomes What are examples? Flip a coin Heads or tails

10 statistical processes 10 Frequentist Probability Defining Terms Sample space Collection of all simple events of experiment Could be population or sample Set notation S = { e 1, e 2, …, e n } where, S  sample space e i  possible simple event (outcome) What is sample space for rolling 1 die? What is sample space for rolling 2 dice?

11 statistical processes 11 Visualizing Sample Space Venn Diagram Venn diagram represents all simple events in sample space Is S 0 part of a larger sample space? S0S0 S 0  all men in VA S1S1 S 1  all >6’ men in VA S2S2 S 2  all men >50 in VA

12 statistical processes 12 Set Terminology Subsets S 0  S 1 S 1 is a subset of S 0 (S 0 is a superset of S 1 ) Every point in S 1 is in S 0 NOTE:S 1 could be the same as S 0 S0  S1S0  S1 S 1 is a strict subset of S 0 Every point in S 1 is in S 0 and S 0  S 1

13 statistical processes 13 Defining Probability p(e i )  probability of e i Likelihood of e i occurring if perform experiment Proportion of times you observe e i Recall frequentist viewpoint in word “size”

14 statistical processes 14 Fundamental Rules Probability If p(e i ) = 0  e i will never occur If p(e i ) = 1.0  e i will occur with certainty Let, E = {e i, …, e j } then, p(E) = p(e i ) + … + p(e j ) Have 2 dice, find p(toss a 7), p(toss an 11)

15 statistical processes 15 Defining More Terms Compound Events Let A  event, B  event A  B is the union of A and B (either A or B or both occur) If C = A  B then A  C, and B  C If A  event you toss 7, B  event you toss 11, and C = A  B What is C Recall E = {e i, …, e j } EventSimple events

16 statistical processes 16 Visualizing Union of Sets Venn Diagrams A B C = A  B A B A B

17 statistical processes 17 Defining More Terms Intersection of Sets S0S0 S 0  all men in VA S1S1 S 1  all >6’ men in VA S2S2 S 2  all men >50 in VA Let C = S 1  S 2 What does C represent??

18 statistical processes 18 Intersection of Sets Dice Example Consider toss of 2 dice, let A = event you toss a 7 B = event you toss an 11 C = A  B Draw Venn Diagram showing C A B A and B are mutually exclusive  A  B =  (the null set)

19 statistical processes 19 Complementarity A Useful Concept Let A be an event then ~A is event that A does not occur ~A is the complement of A ~A read as “not A” also shown as A c, A A c and A read as “the complement of A” p(A) + p(~A) = 1.0 S A ~A

20 statistical processes 20 Conditional Probability Strings Attached Are these likely the same? p(person in VA > 6’ tall) p(person in VA > 6’ tall given person is a man) Former is an unconditional probability Latter is a conditional probability Probability of one event given another event has occurred Formal nomenclature p(A  B)

21 statistical processes 21 Conditional Probability Formula S BA A  B

22 statistical processes 22 Conditional Probabilities Example Problem Study of SPC success at plants A = plant reports success; B = plant reports failure C = plant has formal SPC; D = plant has no formal SPC What are: p(A  C)? p(C)? p(A  C)? p(B  C)?

23 statistical processes 23 Additive Rule of Probability Intuitive Result Additive Rule for Mutually Exclusive Events 1) p(A  B)=0 2) p(A  B) = p(A) + p(B) What if A & B are mutually exclusive? S BA A  B

24 statistical processes 24 Exercise Deck of 52 playing cards What is p(picking a heart or a jack)???

25 statistical processes 25 Exercise Same deck of 52 cards What is p(jack  card is a heart)? What is p(heart  card is a jack)? Your results should make sense

26 statistical processes 26 Multiplicative Rule Recall, conditional probability formula p(A  B) = p(A  B) / p(B) Multiplicative Rule p(A  B) = p(B) p(A  B) = p(A) p(B  A) Remember: Additive rule applies to p(A  B) Multiplicative rule applies to p(A  B)

27 statistical processes 27 Special Case of Conditional Probability: What if the Conditions Do Not Matter? What is p(toss head  previous toss was tail)? p(toss head  previous toss was tail) = p(toss head) Independent events defined as p(A  B) = p(A) p(B  A) = p(B) Multiplicative rule for independent events p(A  B) = p(B) p(A) = p(A) p(B)

28 statistical processes 28 Confirming Independence Do Not Trust Intuition Can Venn Diagrams illustrate independence? No! Unlike mutually exclusive events How to demonstrate A & B are independent? See ifp(A  B) = p(B) p(A) See Examples 3.16 & 3.17, assigned problem 3.24 Not through Venn Diagram Are mutually exclusive events independent? No! p(A  B) = 0  p(B) p(A)

29 statistical processes 29 Counting Rules Counting rules Finding number of simple events in experiment aka Combinatorial Analysis Why would this be important? Most important rules Permutations Combinations

30 statistical processes 30 Permutations Representative Application You are employer 2 open positions, J1 and J2 5 applicants {A, B, C, D, E} for either job How many ways to fill positions??

31 statistical processes 31 Permutations Visualizing Problem Decisions to fill open jobs And so forth. Total of 20 possibilities. Decision tree representation Tool for sequential combinatorial analysis J1J1 A B C D E B J2J2 C D E

32 statistical processes 32 Permutation Formula Is A getting J1 same as A getting J2? Order important Basic distinction of permutation problems Permutation formula N! said as “N factorial” N! = (N)(N-1) … (1) 0! = 1 Multiplicative Rule: Basis of permutation formula

33 statistical processes 33 Permutation Rule More Formal Definition  { e j  j = 1, …, N} Given S N  { e j  j = 1, …, N} S N Select subset of n members from S N Order is important

34 statistical processes 34 Combinations Order Is Not Important Suppose J 1 and J 2 were the same Order not important How would you enumerate combinations? Choose A for J 1 AB, AC, AD, AE Choose B for J 1 BC, BD, BE Choose C for J 1 CD, CE Choose D for J 1 DE A total of 10 combinations!

35 statistical processes 35 Combinations Rule More Formal Definition  { e j  j = 1, …, N} Given S N  { e j  j = 1, …, N} Select subset of n members from S N Order is not important Effectively a sample from S N

36 statistical processes 36 Combinations Rule Different Perspective How many ways can you break up set S N into two subsets: one with n and the other with (N-n) members? S N Set with N members S N Set with N members Subset with n members Subset with n members Subset with (N-n) members Subset with (N-n) members

37 statistical processes 37 Interpreting the Combinations Rule Original set One of the subsetsThe second subset Can you generalize breaking up into > 2 subsets???

38 statistical processes 38 Partitions Rule Breaking Set into k Subsets  { e j  j = 1, …, N} Given S N  { e j  j = 1, …, N} Select k subsets from S N Each subset has n 1, n 2, …, n k members Order is not important Note special case when k=2

39 statistical processes 39 Partitions Rule A Personal Experience Have 55 kids, how many different teams of 11 players each?

40 statistical processes 40 Useful Excel Functions When You Work With Real Data Statistical Special Functions Statistical Special Functions Excel MEAN MEDIAN MODE PERMUT PERCENTILE FACT STDEV STDEVP VAR VARP DEVSQ


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