Lecture5_Spring11

# Lecture5_Spring11 - Elec 210 Lecture 5 Sequential...

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Elec210 Lecture 5 1 Elec 210: Lecture 5 Sequential Experiments e.g., the probability of having k heads when tossing the coin n times e.g., the probability of k heads before a tail comes up Example: Bean Machine Game!

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Elec210 Lecture 5 2 Sequential Experiments Experiments that involve repetitions or multiple participants can often be viewed as a sequence of sub-experiments. The sub-experiments can be identical or non-identical , dependent or independent . The individual sample spaces can be identical or non-identical. If the experiments are identical, the individual sample spaces are identical but not vice versa. Examples: Tossing a coin n times repetition independent identical sub-experiments identical individual sample spaces Checking the number of students who are sleeping in class now multiple participants independent? identical individual sample spaces non-identical sub-experiments Alice and Bob sleep with different probabilities
Elec210 Lecture 5 3 Sample Spaces Formed by Cartesian Products When an experiment consists of performing sub-experiments E 1 , E 2 ,…, E n . The outcome is an n -tuple s = ( s 1 , s 2 ,…. s n ). The sample space can be denoted as the Cartesian product of the individual sample spaces. Example: Toss a coin two times. The sample space is {H,T} × {H,T} = {HH, HT, TH, TT} Example: Pick a random number X from [0,1] and a random number Y from [0.2,0.8]. The sample space is [0,1] × [0.2,0.8] n S S S × × × ....... 2 1 1 0.2 0.8 X Y

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Elec210 Lecture 5 4 Sequences of Independent Sub-Experiments In many cases, the sub-experiments can be assumed to be independent , e.g. a sequence of rolls of a die a sequence of coin flips a sequence of selections from a large collection of resistors. In this case, we can compute the probability of any event by exploiting independence . Let A 1 , A 2 ,…, A n be events such that A k concerns only the outcome of the k th sub-experiment. If the sub-experiments are independent, the events A k are independent and we have: ] [ ].... [ ] [ ] ...... [ 2 1 2 1 n n A P A P A P A A A P = vice versa ?
Elec210 Lecture 5 5 Elec 210: Lecture 5 Sequential Experiments Bernoulli trials Binomial probability law Multinomial probability law Geometric probability law Sequences of dependent experiments Example: Bean Machine Game! Wiki Bernoulli family! Sir Jacob Bernoulli

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Elec210 Lecture 5 6 Bernoulli Trials A Bernoulli trial is an experiment that is performed once with two possible outcomes Example: SUCCESS FAILURE Probability of success: p Probability of failure: q = 1 - p A Bernoulli trial is the simplest experiment Binary sample space: {success, failure}, {1, 0}, {head, tail} Examples Tossing a coin Scoring above the mean in a test Red or black on Roulette wheel
Example 2.37 Suppose that a coin is tossed three times. Assuming the outcomes of the tosses are independent and the probability of heads is always p , the probability for each sequence of heads and tails is

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• Spring '11
• ?
• Probability theory, Binomial distribution, Coin flipping, probability law, Geometric Probability Law, Multinomial Probability Law

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Lecture5_Spring11 - Elec 210 Lecture 5 Sequential...

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