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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 σ μ σ σ μ σ σ σ σ μ σ μ σ σ μ σ σ μ σ σ σ σ - + = - - - - = = - ≤ = 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 μ σ μ σ σ μ σ σ μ σ σ σ σ - + = - - - - = = - ≤ = 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 μ μ ..., , , 2 1 and variances 2 2 2 2 1 ..., , n σ σ 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).
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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 λ pdf of X: f(x) = x e λ λ - , x> 0 where λ > 0 Graph of the pdf of the Exponential Distribution
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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 λ μ 1 ) ( = = X E 2 2 1 ( ) Var X σ λ = =
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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 between failures is less than 550? b.) What is the probability the time interval between failures is between 550 and 700? c.) What is the median time interval between failures?
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d.) What is the probability that it will last more than 550 hours? e.) Given that it has lasted 200 hours, what is the probability that it will last more than an additional 550 hours? This illustrates the lack of memory property of the Exponential Distribution. (The probability that we must wait an additional t unit, given that we have already waited s units, is the same as the probability that we must wait t units from the start.)
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Probability Plots Scientists and engineers often work with data that can
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