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Unformatted text preview: STAT 410 Examples for 09/26/2011 Fall 2011 Normal (Gaussian) Distribution . μ – mean σ – standard deviation 2 , σ μ N ( ) ( ) 2 2 σ μ 2 2 1 σ π = x e x f , ∞ < x < ∞ . Standard Normal Distribution – N ( , 1 ) : μ = 0, σ 2 = 1. Z ~ N ( , 1 ) X ~ N ( μ , σ 2 ) σ μ X Z = X = μ + σ Z ___________________________________________________________________________ EXCEL: ( Z – Standard Normal N ( , 1 ) ) = NORMSDIST( z ) gives Φ ( z ) = P( Z ≤ z ) = NORMSINV( p ) gives z such that P( Z ≤ z ) = p = NORMDIST( x , μ , σ , 1 ) gives P( X ≤ x ), where X is N ( μ , σ 2 ) = NORMDIST( x , μ , σ , ) gives f ( x ), p.d.f. of N ( μ , σ 2 ) = NORMSINV( p , μ , σ ) gives x such that P( X ≤ x ) = p , where X is N ( μ , σ 2 ) ___________________________________________________________________________ 1. Models of the pricing of stock options often make the assumption of a normal distribution. An analyst believes that the price of an Initech stock option varies from day to day according to normal distribution with mean $9.22 and unknown standard deviation. a) The analyst also believes that 77% of the time the price of the option is greater than $7.00. Find the standard deviation of the price of the option. μ = 9.22, σ = ? Know P( X > 7.00 ) = 0.77. Find z such that P( Z > z ) = 0.77. Φ ( z ) = 1 – 0.77 = 0.23 . z = – 0.74 . x = μ + σ ⋅ z . 7.00 = 9.22 + σ ⋅ ( – 0.74 ). σ = $ 3.00 . b) Find the proportion of days when the price of the option is greater than $10.00 ? P( X > 10.00 ) =  > 00 3 22 9 00 10 P . . . Z = P( Z > 0.26 ) = 1 – Φ ( 0.26 ) = 1  0.6026 = 0.3974 . c) Following the famous “buy low, sell high” principle, the analyst recommends buying Initech stock option if the price falls into the lowest 14% of the price distribution, and selling if the price rises into the highest 9% of the distribution. Mr. Statman doesn’t know much about history, doesn’t know much about biology, doesn’t know much about statistics, but he does want to be rich someday. Help Mr. Statman find the price below which he should buy Initech stock option and the price above which he should sell. Need x = ? such that P( X < x ) = 0.14. Find z such that P( Z < z ) = 0.14. The area to the left is 0.14 = Φ ( z ). z = – 1.08 . x = μ + σ ⋅ z . x = 9.22 + 3 ⋅ ( – 1.08 ) = $5.98 . Buy if the price is below $5.98. Need x = ? such that P( X < x ) = 0.09. o Find z such that P( Z < z ) = 0.09. The area to the left is 0.91 = Φ ( z ). z = 1.34 . t x = μ + σ ⋅ z . x = 9.22 + 3 ⋅ ( 1.34 ) = $13.24 . Sell if the price is above $13.24. X ~ N ( μ , σ 2 ) ⇔ M X ( t ) = 2 2 2 σ μ t t e + ....
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This note was uploaded on 10/31/2011 for the course MATH 464 taught by Professor Monrad during the Fall '08 term at University of Illinois, Urbana Champaign.
 Fall '08
 Monrad
 Statistics, Normal Distribution, Probability

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