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Lect12_L6_L7_handout1

# Lect12_L6_L7_handout1 - lecture 12 Continuous Random...

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lecture 12, Continuous Random Variables III Review More Examples Finding z -Points VaR Normal Ap- proximation lecture 12, Continuous Random Variables III Outline 1 Review 2 More Examples 3 Finding z -Points 4 VaR 5 Normal Approximation

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lecture 12, Continuous Random Variables III Review More Examples Finding z -Points 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 z -Points VaR Normal Ap- proximation lecture 12, Continuous Random Variables III Review Properties ( ctd) The curve is symmetric about its center Center=Mean = Median = Mode

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lecture 12, Continuous Random Variables III Review More Examples Finding z -Points 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 z -Points VaR Normal Ap- proximation lecture 12, Continuous Random Variables III Review A Standard Normal Table

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lecture 12, Continuous Random Variables III Review More Examples Finding z -Points 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 z -scores): 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 z -Points 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? 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

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lecture 12, Continuous Random Variables III Review More Examples Finding z -Points VaR Normal Ap- proximation lecture 12, Continuous Random Variables III More Examples Earning Risk An investment: initial cost = \$10M Present value of cash flow X is normally distributed with mean μ = \$12M and standard deviation σ = \$5M Q1. What’s the probability that this investment will make profit?
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