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# normnotes - Normal and Lognormal Distributions...

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Normal and Lognormal Distributions – De…nitions/Notes De…nition 1 A continuous random variable X is said to have a normal distribution with para- meters ¹ and ¾ (or ¹ and ¾ 2 ), where ¡1 < ¹ < 1 and 0 < ¾ , if the pdf of X is: f ( x ; ¹;¾ ) = 1 ¾ p 2 ¼ exp Ã ¡ ( x ¡ ¹ ) 2 2 ¾ 2 ! , for ¡1 < x < 1 . De…nition 2 The normal distribution with parameter values ¹ = 0 and ¾ = 1 is called a standard normal distribution . A random variable that has a standard normal distribution is called a standard normal random variable and will be denoted by Z . The pdf of Z is: f ( x ;0 ; 1) = 1 p 2 ¼ exp Ã ¡ z 2 2 ! , for ¡1 < z < 1 . The cdf of Z is P ( Z · z ) = R z ¡1 f ( y ;0 ; 1) dy , which we will denote by ©( z ) . Notation z ® will denote the value on the measurement axis for which ® of the area under the z curve lies to the right of z ® . Proposition 1 If X has a normal distribution with mean ¹ and standard deviation ¾ , then Z = X ¡ ¹ ¾ is a standard normal random variable.

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## This note was uploaded on 01/08/2012 for the course EXST 4050 taught by Professor Staff during the Fall '10 term at LSU.

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normnotes - Normal and Lognormal Distributions...

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