{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

ContRVs

# ContRVs - Outline The Density Function The Normal Random...

This preview shows pages 1–8. Sign up to view the full content.

Outline The Density Function The Normal Random Variable The Exponential Distribution Transformations Continuous Random Variables Michael Akritas Michael Akritas Continuous Random Variables

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

View Full Document
Outline The Density Function The Normal Random Variable The Exponential Distribution Transformations The Density Function The Normal Random Variable Normal Approximation to the Binomial The Exponential Distribution Transformations Michael Akritas Continuous Random Variables
Outline The Density Function The Normal Random Variable The Exponential Distribution Transformations Deﬁnition X is continuous if there exists a 0 function f ( x ) such that P ( X A ) = Z A f ( x ) dx . This function is called the probability density function , or pdf. Properties: I R -∞ f ( x ) dx = 1 (= P ( -∞ < X < )) . I P ( a X b ) = P ( a < X < b ) = R b a f ( x ) dx . I P ( X = a ) = R a a f ( x ) dx = 0. Read Example 1a, p. 187. Michael Akritas Continuous Random Variables

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

View Full Document
Outline The Density Function The Normal Random Variable The Exponential Distribution Transformations Example X = life time of an indicator light, has pdf f ( x ) = 100 x I ( x > 100) 1. Find P ( X < 150) 2. What is the probability that two of 5 such indicator lights will have to be replaced within the ﬁrst 150 hours? Deﬁnition F ( x ) = P ( X x ) = R x -∞ f ( t ) dt is called the cumulative distribution function. I By the Fundamental Theorem of Calculus, d dx F ( x ) = f ( x ) Michael Akritas Continuous Random Variables
Outline The Density Function The Normal Random Variable The Exponential Distribution Transformations Example X f X ( x ). Find the pdf of Y = 2 X . I μ X = E ( X ) = R -∞ xf X ( x ) dx (Deﬁnition) I E ( g ( X )) = R -∞ g ( x ) f ( x ) dx (To be shown – see next page) I E ( ag 1 ( X ) + bg 2 ( X )) = aE ( g 1 ( X )) + bE ( g 2 ( X )) Example X f X ( x ) = I (0 x 1). Find E ( e X ). Solution: a) Apply above formula with g ( x ) = e x b) Find the pdf, f Y ( y ), of Y = e X and use the deﬁnition of expected value, i.e. E ( Y ) = R -∞ yf Y ( y ) dy . Michael Akritas Continuous Random Variables

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

View Full Document
Outline The Density Function The Normal Random Variable The Exponential Distribution Transformations Lemma If Y is a 0 r.v., then E ( Y ) = R 0 P ( Y > y ) dy Proposition E ( g ( X )) = R -∞ g ( x ) f ( x ) dx. Proof: Assume ﬁrst that g ( x ) 0, for all x. Then, E ( g ( X )) = Z 0 P ( g ( X ) > y ) dy = Z 0 Z x : g ( x ) > y f ( x ) dxdy = Z -∞ Z g ( x ) 0 dyf ( x ) dx = Z -∞ g ( x ) f ( x ) dx For general g write g ( x ) = g ( x ) I ( g ( x ) 0) + g ( x ) I ( g ( x ) < 0) = g + ( x ) - g - ( x ), and use E ( g ( X )) = E ( g + ( X )) - E ( g - ( X )) Michael Akritas Continuous Random Variables
Outline The Density Function The Normal Random Variable The Exponential Distribution Transformations I σ 2 X = Var( X ) = E ( X - μ X ) 2 = E ( X 2 ) - μ 2 X .

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 22

ContRVs - Outline The Density Function The Normal Random...

This preview shows document pages 1 - 8. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online