Lecture11_Spring11

Lecture11_Spring11 - Elec210 Lecture 11 o Expectation of...

Info iconThis preview shows pages 1–10. Sign up to view the full content.

View Full Document Right Arrow Icon
Elec210 Lecture 11 1 Elec210 Lecture 11 o Expectation of Continuous Random Variables o Variance of Continuous Random Variables o Important Continuous Random Variables
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Elec210 Lecture 11 2 Review: Expectation Interpretation The “ average ” value of a random variable if we repeat the underlying experiment a large number of times. The limit of the sample mean as the number repetitions of the experiment increases. Our “ best ” guess of the value of the random variable. (More on this later.) For a discrete RV , However, for a continuous RV , [ ] ( ) where ( ) [ ] kX k X k k k E Xx p x p x P X x  [ ] 0 for all . P x
Background image of page 2
Continuous Elec210 Lecture 11 3
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
From Discrete to Continuous RV Approach: Repeat the experiment for a continuous random variable n times. Divide the real line into small intervals and count the number of times N k ( n ) that the observations fall into each interval. E.g., As n , the relative frequency of the observations falling into the interval approaches the probability of Thus, as 0, the sample mean approaches Elec210 Lecture 11 4   kk xX x  0 () ( ) kX k X n n Xx f n x f x x f x d x         x ) ( ) ( ) ( ) ( ) ( k X k X k X k k x f x F x F n n N n f
Background image of page 4
Expectation for Continuous RVs The expected value or mean of a RV is defined as: The expected value is defined only if the integral converges absolutely. Expected value often denoted by , m or . When X is discrete , the formula reduces to the sum we studied earlier: Elec210 Lecture 11 5  dx x xf X E X ) ( ] [  dt x f x X ) ( | | dx x x x p x X E k k k X  ) ( ) ( ] [ dx x x x x p k k k X  ) ( ) ( ) ( k X k k x p x
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Elec210 Lecture 11 6 Example 4.12: Mean of Uniform Random Variable Pdf of a RV X that is uniformly distributed on the interval [ a , b ] is: The expected value is:  otherwise 0 1 b x a a b x f X  2 2 2 1 2 1 2 2 2 a b a b a b a b a b a b x a b dx a b x X E b a b a b fx () x a 1 b-a ---------------- 2 a b
Background image of page 6
Elec210 Lecture 11 7 Symmetrical PDF Lemma 1 If f X ( x ) is symmetrical around a point m and E [ X ] exists, then E [ X ]= m . Proof Since f X ( x ) is symmetrical around m and ( m -x) is odd symmetric around m:  X E m dx x xf dx x f m dx x f x m X X X  ) ( ) ( ) ( ) ( 0 1 ) (  dt t f X m X E ] [ m - x f X ( x ) x m
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Elec210 Lecture 11 8 The pdf of the Gaussian is symmetric about m : By the previous lemma, E [ X ] = m. Example 4.13: Mean of the Gaussian RV 2 2 2 ) ( 2 1 ) ( m x X e x f
Background image of page 8
Elec210 Lecture 11 9 The pdf of the exponential is Substituting into the definition and integrating by parts, Example 4.14: Mean of the Exponential RV   step unit the is where x u x u e x f x X 1 1 0 0 0 ] [ 0 0 0 0 0
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 10
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 03/27/2011 for the course ELEC 202 taught by Professor ? during the Spring '11 term at HKUST.

Page1 / 37

Lecture11_Spring11 - Elec210 Lecture 11 o Expectation of...

This preview shows document pages 1 - 10. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online