Lecture6_Spring11

Lecture6_Spring11 - Elec 210: Lecture 6 Random Variables...

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Elec210 Lecture 6 1 Elec 210: Lecture 6 Random Variables Equivalent events Discrete Random Variables Probability mass function Sir Isaac Newton and a Famous Probability Problem We are more interested in a numerical attribute, i.e., numbers, of the outcome of the experiment, rather than {tail}, {head}
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Elec210 Lecture 6 2 Random Variables A random variable X is a function that assigns a number to every outcome of an experiment. The function is fixed and deterministic. All randomness in the observed value is due to the underlying experiment ξ 1 IR X ξ () x = S x 1 ξ 2 x 2 Underlying sample space S is called the domain of the random variable. Set of all possible values of X is called the range of the random variable, S X . Range S X Domain
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Elec210 Lecture 6 3 Examples Toss a fair die. Take X = 10 i , where i is the number of dots on the die face Transmit either a 5V pulse or a -5V pulse over a transmission line. Take X = “output voltage sampled at the other end.” Pick a person randomly in the world. Take X = “the height of that person” Record a speech waveform by measuring the voltage at the output of a microphone amplifier. Take X = voltage at time 0.
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Elec210 Lecture 6 4 Example 3.1: Coin Tosses Suppose we toss a fair coin three times. Let X be the number of times heads appears. Sample space: TTT, TTH, THT, THH, HTT, HTH, HHT, HHH Real line: 0123 TTT TTH THT THH HTT HTH HHT HHH ( ) 01121223 X ξ
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Elec210 Lecture 6 5 Example 3.2: A Betting Game Let X be the number of heads in three tosses of a fair coin. The player receives $1 if X = 2, $8 if X = 3 and nothing otherwise. Let Y be the reward to the player. Sample space: TTT, TTH, THT, THH, HTT, HTH, HHT, HHH X: Y: TTT TTH THT THH HTT HTH HHT HHH ( ) 01121223 ( ) 00010118 X Y ξ 0123 4567 8
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Elec210 Lecture 6 6 Elec 210: Lecture 6 Random Variables Equivalent events Discrete Random Variables Probability mass function Sir Isaac Newton and a Famous Probability Problem
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Elec210 Lecture 6 7 Equivalent Event Events on the real line inherit probabilities from the original experiment. The probability of an event is the probability of the equivalent event : This is the set of all outcomes that map to B . These two events are equivalent because if B occurs, then A must also occur and vice versa.
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This note was uploaded on 03/27/2011 for the course ELEC 202 taught by Professor ? during the Spring '11 term at HKUST.

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Lecture6_Spring11 - Elec 210: Lecture 6 Random Variables...

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