# WEEK3 - Notes: The symmetry of normal distribution implies...

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Notes: The symmetry of normal distribution implies that, for any constant a≥0, P(Z≤ -a)= P(Z≥ a) ( ) ( ) ( ) ( ) [1 ( )] 2 ( ) 1 P z Z z z z z z z - ≤ = Φ - Φ - = Φ - - Φ = Φ - 68-95-99.7% Rule: ( ) ( ) ( ) ( 1 1) (1) ( 1) (1) [1 (1)] 2 (1) 1 2 0.8413 1 0.6826 ( 2 2 ) ( 2 2 ) 2 2 ( ) ( 2 2) (2) ( 2) (2) [1 (2)] 2 (2) 1 2 0.9772 1 P X P X X P P Z P X P X X P P Z μ σ μ σ - + = - ≤ - - - = = - ≤ = Φ -Φ - = Φ - -Φ = Φ - = × - = - + = - - - - = = - ≤ = Φ -Φ - = Φ - -Φ = Φ - = × - = 0.9544 ( 3 3 ) ( 3 3 ) 3 3 ( ) ( 3 3) (3) ( 3) (3) [1 (3)] 2 (3) 1 2 0.9987 1 0.9974 P X P X X P P Z - + = - - - - = = - ≤ = Φ -Φ - = Φ - -Φ = Φ - = × - =

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Linear Combinations of Independent Normal Random Variables If X 1 , X 2 , . . ., X n are independent variables from a Normal distribution with means n μ ..., , , 2 1 and variances 2 2 2 2 1 ..., , n σ c 1 , c 2 , . . ,c n are constants then Ex: The molarity of a solute in a solution is defined to be the number of moles of solute per liter of solution. If X is the molarity of a solution of magnesium chloride (MgCl 2 ), and Y is the molarity of a solution of ferric chloride (FeCl 3 ), the molarity of chloride (Cl - ) in a solution made of equal parts of the solutions of MgCl 2 and FeCl 3 is given by M=X+1.5Y. Assume that X has mean 0.125 and standard deviation 0.05, and that Y has mean 0.350 and standard deviation 0.10. Find the mean and standard deviation of M (assuming X and Y are independent).
So, if we add the fact that X and Y are normally distributed, what is the distribution of M? Ex. Assume that X 1 , X 2 , . . ., X n are iid Normal with mean μ and variance σ 2 . 1.) Write X as a linear combination. 2.) What is the E( X )? 3.) What is the variance of X ? 4.) What is the distribution of X ?

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Section 4.7: Ex ponential Distribution Useful in describing the time that elapses before an event occurs (often called waiting time) It is also used to model lifetimes of a component. X is a continuous random variable X~ Exponential ( λ ) denotes that X follows exponential distribution with parameter 0 pdf of X: f(x) = x e - , x> 0 where > 0 Graph of the pdf of the Exponential Distribution
cdf of the Exponential Distribution F(x) = 0 for 0 x x e λ - - 1 for x >0 Mean and Variance of the Exponential Distribution μ 1 ) ( = = X E 2 2 1 ( ) Var X σ = =

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Ex1. An engineer observes that the time before failure for a part of machinery can be modeled as an exponential with a mean of 500 hours. a.) What is the probability that the time interval
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## This note was uploaded on 08/05/2011 for the course STA 3032 taught by Professor Kyung during the Summer '08 term at University of Florida.

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WEEK3 - Notes: The symmetry of normal distribution implies...

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