# 02_22ans - STAT 410 1 Examples for Spring 2008 Let X be...

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STAT 410 Examples for 02/22/2008 Spring 2008 1. Let X be normally distributed with mean μ and standard deviation σ . Find the moment-generating function of X , M X ( t ) . M X ( t ) = E ( e t X ) = ( ) ° - - - dx x x t e e 2 2 ° ± 2 1 2 ° π = ( ) ° - - + dz z z t e e 2 2 2 1 ° ± π = ( ) ° - - - + dz t z t t e e 2 2 2 2 2 ° 2 1 ° ± π = 2 2 2 ° ± t t e + , since ( ) 2 2 ° 2 1 t z e - - π is the probability density function of a N ( σ t , 1 ) random variable. 2. Models of the pricing of stock options often make the assumption of a normal distribution. An investor believes that the price of an Burger Queen stock option is a normally distributed random variable with mean \$18 and standard deviation \$3. He also believes that the price of an Dairy King stock option is a normally distributed random variable with mean \$14 and standard deviation \$2. Assume the stock options of these two companies are independent. The investor buys 8 shares of Burger Queen stock option and 9 shares of Dairy King stock option. What is the probability that the value of this portfolio will exceed \$300? BQ has Normal distribution, μ BQ = \$18, σ BQ = \$3. DK has Normal distribution, μ DK = \$14, σ DK = \$2.

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