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Notes_prob - Notes on Probability and Random Variables...

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Notes on Probability and Random Variables Optimal Estimation (EML 6934, Section 6385) Fall 2009 University of Florida, Mechanical and Aerospace Engineering Prabir Barooah * 1 Probability 1.1 Random Experiment Everything is based on a “random experiment”, which is a mathematical model for some unpredictable phenomenon. All possible outcomes of the random experiment are collected in a set Ω, which is called the sample space . Examples: 1. A die is thrown : possible outcomes are “1 dot”, “2 dots”, ..., “6 dots”. Then Ω = { 1 dot , 2 dots , . . . , 6 dots } 2. A coin is tossed twice. Then Ω = { HH, HT, T H, T T } . 3. System identification experiment described in class: Ω consists of all possible values of the paur a d , ˆ b d ). An event is a collection of such outcomes. Examples: 1. (in the die toss experiment) E Δ = { 1 , 2 , 4 , 6 } . This is the event that an even number of dots turn up. 2. (in the system identification experiment) E Δ = { ˆ a d < 1 } . (What’s this event?) In general, and event E is a subset of the sample space, i.e., E Ω. Based on the sample space Ω, a larger set F is defined: F = { E | E Ω , E is “measurable” } which is called a σ -field or σ -algebra . We will not go into what a measurable set is. For the purpose of this course, you can take it that all sets are measurable, so that every subset of Ω is an element of the σ -algebra F . Think of F as the set of all events. F contains Ω and φ , the sample space and the empty set, among other events. When Ω is thought of as an event (an element of F ), it is called the sure event. φ is called the impossible event. There are several ways of thinking of probability. * Please email me at [email protected] if you find any typos. 1
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1. Probability as the relative frequency of occurrence : do the underlying random experiment a very large number of times. Probability of an event is the ratio between the number of times the event occurs to the total number of experiments. 2. Probability as the ratio of favorable to all possible outcomes: Example: say the event E = an even number of dots appear when a die is thrown. The favorable outcomes, for which the event is said to occur, are 2 , 4 , 6, whereas all the possible outcomes are 1 , . . . , 6. Then, probability of E is 3 / 6, i.e., 1 / 2. Note that all the outcomes are assumed to be equally likely for this calculation to make sense. 3. Modern, or axiomatic, definition of probability: A probability P is a function that assigns a number between 0 and 1 to every event, i.e., to every element of the σ -field F . Therefore, probability is a function P : F [0 , 1] that satisfies the following axioms: P ( E ) 0 for every E F . P (Ω) = 1. If A B = φ , then P ( A B ) = P ( A ) + P ( B ) The triplet (Ω , F, P ) is called a probability space . Two events A and B are called mutually exclusive if A B = φ . 1.1.1 Operations on sets A B Δ = { ω Ω | ω A, ω B } = “ A and B ” (both the events A and B occur) A B Δ = { ω Ω | ω A or ω B } = “ A or B ” (either A occurs, or B occurs, or both occur) A c Δ = { ω Ω | ω / A } = “not A” (A does not occur) A - B Δ = { ω Ω | ω A, ω / B } = “ A
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