Chen iecuhk engg2430c lecture 5 7 21 random variable

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Unformatted text preview: 430C lecture 5 6 / 21 Random Variable: Motivation M. Chen ([email protected]) ENGG2430C lecture 5 7 / 21 Random Variable: Definition A very “misleading” name: Random variable is not random, it is deterministic Random variable is not a variable, it is a function Definition: A random variable is a deterministic function from the sample space Ω to the real numbers Discrete or continuous Functions of random variables are also random variables: Y = f (X ) is still a function from Ω to real numbers M. Chen ([email protected]) ENGG2430C lecture 5 8 / 21 Examples Define random variable X and Y for the disease testing experiment: X (ω ) = Y (ω ) = 1, ω = having the disease 0, ω = healthy 1, ω = test is positive 0, ω = test is negative Events can be expressed using random variables: {having the disease}:{ω : X (ω ) = 1} = {X = 1} {having the disease and the test is positive}:{X = 1, Y = 1}. M. Chen ([email protected]) ENGG2430C lecture 5 9 / 21 Examples Experiment: Flip a fair coin and obtain two possible outcomes: HEAD or TAIL Define a Bernoulli random variable X as X (ω ) = 1, ω = outcome is ’HEAD’ 0, ω = outcome is ’TAIL’ Experiment: Flip a fair coin independently until we obtain the first HEAD Define a Geometric random variable Y as 1, ω = outcome is ’H’ 2, ω = outcome is ’TH’ Y (ω ) = 3, ω = outcome is ’TTH’ . . . . ., . M. Chen ([email protected]) ENGG2430C lecture 5 10 / 21 Probability Mass Function (PMF) Probability distribution: assign probability to events described by random variables. Discrete random variable: probability mass function (PMF) Continuous random variable: probability density function (PDF) PMF of a discrete random variable X : pX (x ) Examples: P ({X = x }) Define a r.v. X as the outcome of a coin toss (assuming P (H ) = p > 0) pX (1) = p , pX (0) = 1 − p Define a r.v. Y as the number of independent coin tosses until we receive the first head pY (k ) = P (Y = k ) (all but the last toss m...
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This document was uploaded on 03/31/2014.

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