Normal[1] - Normal Distribution A continuous random...

Info iconThis preview shows pages 1–9. Sign up to view the full content.

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon

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

View Full DocumentRight Arrow Icon

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

View Full DocumentRight Arrow Icon

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

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

Unformatted text preview: Normal Distribution A continuous random variable X has a normal distribution and it is referred to as a normal random variable if its probability density is given by 2 2 2 ( ) /2 1 ( ; , ) for 2 x f x e x μ σ μ σ πσ- - = -∞< < ∞- < < > 0 μ σ ∞ ∞ where and are the parameters of the normal distribution. The normal distribution is also known as Gaussian distribution because the Gaussian function is defined as 2 2 / ) ( ) ( c b x e a x f-- = Normal probability density function for selected values of the parameter μ and σ 2 2 ( ; , ) xf x dx μ σ ∞-∞ = ∫ mean 2 2 ( ) / 2 1 2 x x e dx μ σ πσ ∞---∞ = ∫ 2 2 ( ) / 2 1 2 x x e dx μ σ σ π ∞---∞ = ∫ 2 2 ( ) / 2 1 2 x x e dx μ σ μ μ σ σ π ∞---∞- = + ∫ 2 / 2 2 1 ( ; , ) . 2 u ue du f x dx σ μ μ σ π ∞ ∞--∞-∞ = + ∫ ∫ μ = 2 2 2 2 2 ( ) / 2 2 2 2 / 2 2 / 2 1 Var(X) ( ) 2 2 2 2 x u u x e dx u e du u e du μ σ μ πσ σ σ π π ∞- --∞ ∞ ∞- --∞ = - = = ∫ ∫ ∫ ( 29 2 2 / 2 2 2 / 2 1. 2 . u u ue du du u ue du σ π ∞ ∞- - =- ∫ ∫ ∫ 2 2 2 2 / 2 / 2 2 2 2 / 2 2 2 u u u e du e du ue σ σ π π ∞ ∞ ∞-- - = + = - ∫ ∫ 2 σ = Moment Generating Function – ( 29 2 2 ( ) / 2 1 ( ) 2 tX tx x X M t E e e e dx μ σ πσ ∞---∞ = = ∫ 2 2 2 1 ( ) 2 2 1 2 x tx e dx μ σ σ πσ ∞ --- -∞ = ∫ 2 2 2 2 1 ( ) 2 ( ) 2 2 1 2 x t x t e dx μ σ μ σ μ σ πσ ∞ ----- -∞ = ∫ 2 2 4 2 4 2 2 2 1 ( ) 2 ( ) 2 2 1 2 x t x t t t e dx μ σ μ σ σ σ μ σ πσ ∞ ---- +-- -∞ = ∫ 2 2 2 2 4 2 2 2 2 2 2 2 2 2 1 ( ) 2 ( ) 2 2 1 ( ) 2 2 2 1 2 1 2 t x t x t t t x t t t t e e dx e e dx e σ μ σ μ σ μ σ σ μ σ μ σ σ μ πσ πσ ∞ ---- + + -∞ ∞ --- + -∞ + = = = ∫ ∫ 2 2 2 ( ) t t X M t e σ μ + = ( ) E X μ = 2 ( ) Var X σ = Standard Normal Distribution – A normal random variable with mean 0 and variance 1 is called standard normal random variable. is called standard normal random variable....
View Full Document

This note was uploaded on 02/23/2011 for the course MATH 112 taught by Professor Ritadubey during the Spring '11 term at Amity University.

Page1 / 24

Normal[1] - Normal Distribution A continuous random...

This preview shows document pages 1 - 9. Sign up to view the full document.

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