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# 419 Class Note for STAT 416 at PSU

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COURSE
PROF.
No professor available
TYPE
Class Notes
PAGES
7
WORDS
KARMA
25 ?

## Popular in Department

This 7 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at Pennsylvania State University taught by a professor in Fall. Since its upload, it has received 26 views.

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Date Created: 02/06/15
Sample Space and Events 0 Sample space The set of all possible outcomes 1 Roll 21 die 12 3456 2 Flip a coin twice H H H T T H T 0 Event A subset of the sample space 1 Roll 21 die the outcome is even 2 4 6 2 Flip a coin twice and the two results are different H T T H 0 Set operations 1 Union E UE an outcome is in EUF if it is either in E or in F 2 Intersection E H F EF an outcome is in EF if it is in both E and F 3 Mutually exclusive E and F are mutually exclu sive if EF gb a Roll a die E1 outcome is below 3 1 2 E2 outcome is above 4 5 6 E1 E2 4 Complement EC outcome that is not in E Probabilities 0 Sample space S event E o The probability of event E is a number PE assigned to E that satis es the following conditions 1 0 g PE g 1 2 PS 1 3 For any sequence of events E1 E2 which are mutually exclusive that is EnE b for any n y ma PUO1En 2201 0 Compute Probabilities 1 Roll a fair die 6 equally likely outcomes 1 2 3 4 5 6 P1 16 P2 16 P6 16 2 E the outcome is even PE PE 136247 6 P2P4P6 12 0 Properties 1 PE PEC l 2 Any event E and F may not be mutually exclu sive PE U F PE PF PEF Proof LetA EFB E ACF A Then EBUAFCUAEUFBUCUA Note A B C are mutually exclusive PE U F PB PC PM 133 PUD 130 PUD PM PE PF PEF Conditional Probability 0 Given one event has occurred what is the probability that another event occurs Let F be given then the conditional probability of E is PE F 0 Example Flip a coin twice S Hv H H7 T T7 H T7 F the rst ip is H F HH H PF 12 E the two ips are not both H E HT T T T PE 34 If F occurs in order for E to occur the second ip has to be T Since the coin is fair PEF 12 0 De nition PEF PF Example the above coin ip setup PltEF PltHT 14 PE FPF PF 12 HEW 12 Independent Events 0 De nition Two events E and F are independent if PEF PEPF o Equivalently PElF PE PFlE PF o Distinguish independent and mutually exclusive Independent PEF E and F mutually exclusive EF 5 PEF 0 PE U F PE PF 0 Extension to n events E1 E2 En are independent if PE1E2E3 En H PE 11 0 Independent trials A sequence of experiments with results being either a success or a failure and the experiments are independent Bayes Formula 0 Total probability formula Suppose events F1 F2 Fn are mutually exclusive and U1F S Given any event E we have PE PFZE PEPE Fl 0 Bayes formula Suppose events F1 F2 Fn are mutually exclusive and U1F S Given any event E i pEPE Fi Pm E 21PFjPE Fj

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