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...
View
Full
Document
This document was uploaded on 03/31/2014.
 Spring '14

Click to edit the document details