Normal[1]

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

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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....
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## 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.

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

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