Lecture 12-2007

# Lecture 12-2007 - Normal Random Variables Lecture XII...

This preview shows pages 1–6. Sign up to view the full content.

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

View Full Document

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

View Full Document

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.

Unformatted text preview: Normal Random Variables Lecture XII Univariate Normal Distribution. • Definition 5.2.1. The normal density is given by • When X has the above density, we write symbolically X ~ N ( μ , σ 2 ). ( 29 ( 29 , 2 1 exp 2 1 2 2 ∞ < < ∞- -- = σ σ μ σ π x x x f Mean and Variance of Normal Distribution • Theorem 5.2.1. Let X be distributed N ( μ , σ 2 ). Then E [ X ]= μ and V [ X ]= σ 2 [ ] ( 29 ∫ ∞ ∞- -- = dx x x X E 2 2 2 2 1 exp 2 1 σ μ σ π dz dx z x x z σ μ σ σ μ = + = ⇒- = [ ] ( 29 ∫ ∫ ∫ ∞ ∞- ∞ ∞- ∞ ∞- - + - = - + = dz z dz z z dz z z X E 2 2 2 2 2 1 exp 2 1 2 1 exp 2 1 2 1 exp 2 1 π μ σ σ π μ σ σ π = -- = - = - ∞ ∞- ∞ ∞- ∞ ∞- ∫ ∫ 2 1 exp 2 1 exp 2 1 exp 2 1 2 2 2 2 z C dz z z C dz z z σ σ π [ ] ( 29 ( 29 ( 29 ∫...
View Full Document

{[ snackBarMessage ]}

### Page1 / 15

Lecture 12-2007 - Normal Random Variables Lecture XII...

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online