351classnotes1.5-1.6

# 351classnotes1.5-1.6 - Section 1.5 Other Useful Continuous...

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Unformatted text preview: Section 1.5 Other Useful Continuous Distributions . Many data sets in practice have histograms which have shapes unimodal, skewed, rises to maximum and then decreases. Some distributions having these properties are: 1. Lognormal Distribution Definition : The variable x is said to said to follow LN ( μ, σ ) if its density is ( 29 , 2 1 2 ) ln( 2 2 1-- x e x x μ σ π ) , | ( σ μ x f = otherwise , . 1 Facts: (i) Putting y = ln (x), it is easy to check ∫ ∞ ∞- = 1, ) , | ( dy y f σ μ for all μ ε |R, σ > 0. (ii) If x ~ L N ( μ, σ ), then y = ln (x) ~ N ( μ, σ ). This fact is used to compute proportions. Example 1. The variable x = modulus of elasticity (MOE) of Wood Joist Floor Systems with Creep is observed, in a study, to follow LN (.375, .25). Find (i) Proportion of systems with x < 3. (ii) Proportion of systems with 1 < x < 2. (iii) For what value of c, only 1% of systems have x (MOE) > c. That is, c is nothing but the 99 th percentile) Solution . (i) Proportion of systems with x < 3. ⇒ y = ln (x) < ln (3) = 1.098 Now y ~ N (.375, .25) Therefore, y < 1.0908 ⇔ .25 .375- 1.098 25 . 375 . <- = y z i.e., z ~ N (0, 1) and Propo. of z < 2.88 From normal tables, proportion of (z < 2.88) = .998. 2 Hence, prop. of systems with x < 3 = .998 (or 99.8%). (ii) Similarly, Prop. of systems with 1 < x < 2 ) 2 ( ln ) ln( ) 1 ( ln < < ⇔ x ) 2 ( ln ) 1 ( ln < < ⇔ y 25 . 375 . ) 2 ln( 25 . 375 . 25 . 375 . ) 1 ln(- <- <- ⇔ y z ~ n (0,1); proportion of z with -1.512 < z < 1.272 Prop. of z < 1.272 = .8984 (see tables) Prof. of z < -1.512 = .0653------- Prop. of z : -1.512 < z < 1.272: .8331 (subtract) Prop. of x with 1 < x < 2 = 83.3% (iii) Need to find c 220d prof x > c = .01 ⇔ prop. of x with x ≤ c = .99 Therefore, find c 220d .99 25 . 375 . ) ln( 25 . 375 . ) ln( =- ≤- c x i.e., c is 220d proportion of 99 . 25 . 375 . ) ln( z = =- ≤ c From tables, 99 th percentile of z is 2.33. 3 Therefore, 2.33 25 . 375 . ) ln( =- c Solving ln ( c ) = .9575 ⇒ c = 2.605. 2. Weibull Distribution: W (α , β) This is an extension of exponential distribution. Definition : A continuous variable x follows W (α , β) if its density is , 1- - x x e x α α β α β α ) , | ( β α x f = elsewhere. , (i) Different values of α and β give positively and negatively skewed distributions. 4 Some graphs of graphs of Weibull distributions are: 5 (ii) Prop. of x with x < t ∫ = t dx x f ) , | ( β α ∫- - = t x dx x e 1 α β α α β α ∫ - = α β t y dy e ....
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351classnotes1.5-1.6 - Section 1.5 Other Useful Continuous...

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