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Unformatted text preview: lecture 12, Continuous Random Variables III Review More Examples Finding zPoints VaR Normal Ap proximation lecture 12, Continuous Random Variables III Outline 1 Review 2 More Examples 3 Finding zPoints 4 VaR 5 Normal Approximation lecture 12, Continuous Random Variables III Review More Examples Finding zPoints VaR Normal Ap proximation lecture 12, Continuous Random Variables III Review Normal Probability Distribution The normal probability distribution with mean μ and standard deviation σ , denoted by N ( μ,σ 2 ) , is f ( x ) = 1 √ 2 πσ 2 e ( x μ ) 2 2 σ 2 for all values x on the real number line μ is the mean and σ is the standard deviation π ≈ 3 . 14159, and e ≈ 2 . 71828 is the base of natural logarithms The normal distribution with mean 0 and standard deviation 1 is called the standard normal distribution . lecture 12, Continuous Random Variables III Review More Examples Finding zPoints VaR Normal Ap proximation lecture 12, Continuous Random Variables III Review Properties ( ctd) The curve is symmetric about its center • Center=Mean = Median = Mode lecture 12, Continuous Random Variables III Review More Examples Finding zPoints VaR Normal Ap proximation lecture 12, Continuous Random Variables III Review Properties ( ctd): Position and Shape • The mean μ positions the peak of the normal curve over the real axis • The sd σ measures the width or spread of the normal curve lecture 12, Continuous Random Variables III Review More Examples Finding zPoints VaR Normal Ap proximation lecture 12, Continuous Random Variables III Review A Standard Normal Table lecture 12, Continuous Random Variables III Review More Examples Finding zPoints VaR Normal Ap proximation lecture 12, Continuous Random Variables III Review Finding Normal Probabilities Given a normal rv X with mean μ and standard deviation σ . Want to find its probabilities. • E.g., want to find P ( a ≤ X ≤ b ) • Calculate the corresponding z values (also called zscores): z = x μ σ ( ⇔ x = σ z + μ ) and restate the problem in terms of these z values • P ( a ≤ X ≤ b ) = P a μ σ ≤ Z ≤ b μ σ . • Find the required areas under the standard normal curve by using the standard normal table It is always useful to draw a picture showing the required areas before using the normal table lecture 12, Continuous Random Variables III Review More Examples Finding zPoints VaR Normal Ap proximation lecture 12, Continuous Random Variables III More Examples Present Value Suppose you were offered a choice between $100 today or $100 in one year, which one would you choose? • Because you can invest your money in a bank account that will return you interest, and because of inflation, $100 today is worth more than $100 in one year • In other words, $100 in one year is worth less than $100 today The value that $100 in one year is worth today is called its present value (PV) Present value (PV) is the discounted value of a future payment lecture 12,...
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This note was uploaded on 12/30/2009 for the course ISOM ISOM111 taught by Professor Anthonychan during the Fall '09 term at HKUST.
 Fall '09
 AnthonyChan

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