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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   = < <  < < > 0 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 and 2 2 ( ; , ) xf x dx  = mean 2 2 ( ) / 2 1 2 x x e dx  = 2 2 ( ) / 2 1 2 x x e dx  = 2 2 ( ) / 2 1 2 x x e dx  = + 2 / 2 2 1 ( ; , ) . 2 u ue du f x dx  = + = 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     =  = = ( 29 2 2 / 2 2 2 / 2 1. 2 . u u ue du du u ue du   = 2 2 2 2 / 2 / 2 2 2 2 / 2 2 2 u u u e du e du ue   = + =  2 = Moment Generating Function ( 29 2 2 ( ) / 2 1 ( ) 2 tX tx x X M t E e e e dx  = = 2 2 2 1 ( ) 2 2 1 2 x tx e dx   = 2 2 2 2 1 ( ) 2 ( ) 2 2 1 2 x t x t e dx   = 2 2 4 2 4 2 2 2 1 ( ) 2 ( ) 2 2 1 2 x t x t t t e dx  +  = 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  + +   +  + = = = 2 2 2 ( ) t t X M t e + = ( ) E X = 2 ( ) Var X = 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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 Spring '11
 ritadubey
 Normal Distribution, Probability

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