Math 489/889 Stochastic Processes and Advanced Mathematical Finance Final Exam Solutions Steve Dunbar Tuesday, December 14, 2010 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 1" layout="Heading 1">1</Text-field> ``Short Answer'', use a single sentence or`True or False'' and if false, give a reason why it is false in a single sentence. (If false, 1 point for the answer, 4 pointsfor the reason.)
<Text-field style="Heading 2" layout="Heading 2">(a)</Text-field> Short Answer: Why is Geometric Brownian Motion a better model of the stock market than Brownian motion with drift, where the drift parameter is the rate \134( r \134) of market growth? Because Geometric Brownian motion is always positive, while there is a positive probability (albeit small) that Brownian Motion with drift can take on negative values.
<Text-field style="Heading 2" layout="Heading 2">(b)</Text-field> True or False: The Black-Scholes pricing equation is based on the model that the underlying stock price follows a Brownian Motion. False, the underlying stock price is assumed to follow a Geometric Brownian motion, a different (but related) stochastic process.
<Text-field style="Heading 2" layout="Heading 2">(c)</Text-field> The Black-Scholes pricing equation values an option by taking the present value of the expected return on the option. False, the Black Scholes equation prices an option by finding an equivalent dynamic portfolio in the underlying security and a bond and then uses the principle of no-arbitrage to find the equivalent price.
<Text-field style="Heading 2" layout="Heading 2">(d)</Text-field> True or False: The closed form solution of the partial differential equation that we call the Black-Scholes formula represents the final word in financial theory. False, it is the starting point and initial ground-breaking idea that stimulated the whole area of mathematical finance.
<Text-field style="Heading 2" layout="Heading 2">(e)</Text-field> True or False: The volatility of a stock price can be estimated from the Black-Scholes Formula if the option values are known from the market. True, that is called the implied volatility.
<Text-field style="Heading 2" layout="Heading 2">(f)</Text-field> True or False: European puts cannot be valued by solving the Black-Scholes equation, only European calls can be valued by solving the Black-Scholes equation. False, they can by solving a different terminal value problem, one that we did not choose to solve, but could have.
<Text-field style="Heading 2" layout="Heading 2">(g)</Text-field> Short answer: What mathematical property of the Black-Scholes equation allows you to write the formula for the value of a strap (a portfolio consisting of one put and two calls, all with the same strike price) in terms of the value for a call and a put other solutions? The Black-Scholes equation is linear, so that the linear combinations of solutions is again a solution.
<Text-field style="Heading 1" layout="Heading 1">2</Text-field> What is the price of a European put option on a non-dividend-paying when the stock price is \134\$69, the strike price is \134\$70, the risk-free interest rate is 5\134% per year (continuously compounded), the volatility is 35\134% per year, and the time to maturity is 6 months. QyQtSSV3aXRoRzYiNiNJKGZpbmFuY2VHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiUiIiI= LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder Ny5JLWFtb3J0aXphdGlvbkc2IkkoYW5udWl0eUdGJEktYmxhY2tzY2hvbGVzR0YkSSpjYXNoZmxvd3NHRiRJLmVmZmVjdGl2ZXJhdGVHRiRJLGZ1dHVyZXZhbHVlR0YkSS9ncm93aW5nYW5udWl0eUdGJEkyZ3Jvd2luZ3BlcnBldHVpdHlHRiRJLGxldmVsY291cG9uR0YkSStwZXJwZXR1aXR5R0YkSS1wcmVzZW50dmFsdWVHRiRJMHlpZWxkdG9tYXR1cml0eUdGJA== QyQtSSV3aXRoRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJK1N0YXRpc3RpY3NHRiUhIiI= %; Qyw+SSJTRzYiIiNwIiIiPkkiS0dGJSIjcUYnPkkickdGJSQiIiYhIiNGJz5JJnNpZ21hR0YlJCIjTkYvRic+SShUbWludXN0R0YlJEYuISIiRic= LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder IiNw IiNx JCIiJiEiIw== JCIjTiEiIw== JCIiJiEiIg== QyQ+SStjYWxsT3B0aW9uRzYiLUkmZXZhbGZHJSpwcm90ZWN0ZWRHNiMtSS1ibGFja3NjaG9sZXNHRiU2J0kiU0dGJUkiS0dGJUkickdGJUkoVG1pbnVzdEdGJUkmc2lnbWFHRiUiIiI= LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder JCIrLlFySHIhIio= Use Put-Call Parity to determine the putOption value: QyQ+SSpwdXRPcHRpb25HNiIsKEkrY2FsbE9wdGlvbkdGJSIiIiomSSJLR0YlRigtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IywkKiZJInJHRiVGKEkoVG1pbnVzdEdGJUYoISIiRihGKEkiU0dGJUY1Rig= LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder JCIqazI5UychIik= JSFH Now value the Put Option directly with the Black-Scholes formula for a Put: QyQ+SSNkMUc2Ii1JJmV2YWxmRyUqcHJvdGVjdGVkRzYjKigsJi1JJGxvZ0c2JEYoSShfc3lzbGliR0YlNiMqJkkiU0dGJSIiIkkiS0dGJSEiIkYzKiYsJkkickdGJUYzKiRJJnNpZ21hR0YlIiIjI0YzRjtGM0koVG1pbnVzdEdGJUYzRjNGM0Y6RjUtSSVzcXJ0R0YlNiNGPUY1RjM= LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder JCIrQG0+bTshIzU= PkkjZDJHNiItSSZldmFsZkclKnByb3RlY3RlZEc2IyooLCYtSSRsb2dHNiRGJ0koX3N5c2xpYkdGJDYjKiZJIlNHRiQiIiJJIktHRiQhIiJGMiomLCZJInJHRiRGMiokSSZzaWdtYUdGJCIiIyNGNEY6RjJJKFRtaW51c3RHRiRGMkYyRjJGOUY0LUklc3FydEdGLTYjRjxGNA== LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder JCErSzZ4JzMpISM2 QyQ+SShjaGVja2QyRzYiLUkmZXZhbGZHJSpwcm90ZWN0ZWRHNiMsJkkjZDFHRiUiIiIqJkkmc2lnbWFHRiVGLC1JJXNxcnRHNiRGKEkoX3N5c2xpYkdGJTYjSShUbWludXN0R0YlRiwhIiJGLA== LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder JCEqOHJuMykhIzU= QyY+SSJaRzYiLUkvUmFuZG9tVmFyaWFibGVHRiU2Iy1JJ05vcm1hbEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYkIiIhIiIiRjA+SSZQaGlkMUdGJS1JJENERkdGJTYkRiQsJEkjZDFHRiUhIiJGMA== LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder SSRfUjBHNiUlKnByb3RlY3RlZEdJL1JhbmRvbVZhcmlhYmxlRzYiSTpfUHJvYmFiaWxpdHlEaXN0cmlidXRpb24wR0Ym JCIzYlUqWyM0bU1RViEjPQ== QyQ+SSZQaGlkMkc2Ii1JJENERkdGJTYkSSJaR0YlLCRJI2QyR0YlISIiIiIi LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder JCIzPjk2azJVRUFgISM9 QyQ+SSJDRzYiLCYqJkkiU0dGJSIiIkkmUGhpZDFHRiVGKSEiIiooSSJLR0YlRiktSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IywkKiZJInJHRiVGKUkoVG1pbnVzdEdGJUYpRitGKUkmUGhpZDJHRiVGKUYpRik= LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder JCIqbDI5UychIik= JSFHJSFH
<Text-field style="Heading 1" layout="Heading 1">3</Text-field> Use the put-call parity relationship to derive the relationship between (a) The Delta of European call and the Delta of European put. (The Delta of an option is the rate of change of option value with respect to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. (b) The Theta of European call and a European put. (The Theta of an option is the rate of change of option value with respect to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIi5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTEY5 Show your complete work. 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY7LUkjbWlHRiQ2JVEucHV0Q2FsbFBhcml0eUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSomY29sb25lcTtGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjc3Nzc3OGVtRicvRk5GU0Y1LUYsNiVRIkNGJ0YvRjItSShtZmVuY2VkR0YkNiQtRiM2Jy1GLDYlUSJTRidGL0YyLUY2Ni1RIixGJ0Y5RjsvRj9GMUZARkJGREZGRkhGSi9GTlEsMC4zMzMzMzMzZW1GJy1GLDYlUSJ0RidGL0YyLyUrZXhlY3V0YWJsZUdGPUY5RjlGNS1GNjYtUSomdW1pbnVzMDtGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjIyMjIyMmVtRicvRk5GaW9GNS1GLDYlUSJQRidGL0YyRlhGNS1GNjYtUSI9RidGOUY7Rj5GQEZCRkRGRkZIRlJGVEY1RmduRjVGZW9GNS1GLDYlUSJLRidGL0YyLUY2Ni1RJyZzZG90O0YnRjlGO0Y+RkBGQkZERkZGSEZKRk0tRiw2JVEkZXhwRicvRjBGPUY5LUZZNiQtRiM2KEZlby1GLDYlUSJyRidGL0YyRmRwLUZZNiQtRiM2Jy1GLDYlUSJURidGL0YyLUY2Ni1RKCZtaW51cztGJ0Y5RjtGPkZARkJGREZGRkhGaG9Gam9GYG9GY29GOUY5RmNvRjlGOS1GNjYtUSI7RidGOUY7Rl1vRkBGQkZERkZGSEZKRlRGY29GOQ== LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LywmLUkiQ0c2IjYkSSJTR0YmSSJ0R0YmIiIiLUkiUEdGJkYnISIiLCZGKEYqKiZJIktHRiZGKi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJjYjLCQqJkkickdGJkYqLCZJIlRHRiZGKkYpRi1GKkYtRipGLQ== JSFH
<Text-field style="Heading 2" layout="Heading 2">(a)</Text-field> 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LywmLUklZGlmZkclKnByb3RlY3RlZEc2JC1JIkNHNiI2JEkiU0dGKkkidEdGKkYsIiIiLUYlNiQtSSJQR0YqRitGLCEiIkYu JSFH
<Text-field style="Heading 2" layout="Heading 2">(b)</Text-field> 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LywmLUklZGlmZkclKnByb3RlY3RlZEc2JC1JIkNHNiI2JEkiU0dGKkkidEdGKkYtIiIiLUYlNiQtSSJQR0YqRitGLSEiIiwkKihJIktHRipGLkkickdGKkYuLUkkZXhwRzYkRiZJKF9zeXNsaWJHRio2IywkKiZGN0YuLCZJIlRHRipGLkYtRjNGLkYzRi5GMw== JSFH
<Text-field style="Heading 1" layout="Heading 1">4</Text-field> Find a numerical approximation at 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 to the solution of the Stochastic Differential Equation: dX = (1 - X) dt + dW; X(0) = 0.5 Remark: With some general parameters, this stochastic differential equation is a model of a ``mean-reverting process'' called the Ornstein-Uhlenbeck process, a useful model in physics and mathematics.) Use LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVEjZHRGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSI9RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlMtSSNtbkdGJDYkUSQwLjJGJ0ZARkBGK0ZARitGQA==, and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVEiTkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUy1JI21uR0YkNiRRJDEwMEYnRkBGQEYrRkBGK0ZA in the table of net totals of randomly generated coin flips below. NiNRJHRfajYi NiNRJFhfajYi NiNRKCgxLVhfaik2Ig== NiNRImo2Ig== NiMiIiE= NiMiIiI= NiMiIiM= NiMiIiQ= NiMiIiU= NiMiIiY= NiMkIiIhIiIh NiMkIiIjISIi NiMkIiIlISIi NiMkIiInISIi NiMkIiIpISIi NiMkIiM1ISIi NiMkIiImISIi NiNRI2RXNiI= NiMkIiImISIi NiNRKygxLVhfaikqZHQ2Ig== NiMkIiM1ISIj NiMkIiIjISIi NiMkISIpISIi NiMkISIlISIi NiMkIiIhIiIh NiMkISIjISIi NiMkIiNJISIj NiNRLigxLVhfaikqZHQrZFc2Ig== NiNRKFhffGZyaisxfGhyNiI= NiMkIiMhKSEiIw== NiMkIiMhKSEiIw== NiMkIiM/ISIj NiMkIiNTISIk NiMkISRnKCEiJA== NiMkIiNTISIk NiMkIiNTISIk NiMkIiRnKiEiJA== NiMkIiU/PiEiJQ== NiMkISUhMyMhIiU= NiMkISUhbyIhIiU= NiMkISUhbyIhIiU= NiMkIiYhbzYhIiU= NiMkIiZnTCMhIiY= NiMkIiVnbCEiJg== NiMkIiVnbCEiJg== NiMkIiZTTSohIiY= NiMkIichKW89ISIn NiMkISY/SiIhIic= NiMkIiZnTCMhIiY= NiMkIiYhW18hIic= NiMkIiYhW18hIic= JSFH
<Text-field style="Heading 1" layout="Heading 1">5</Text-field> A company's cash position, measured in millions of dollars, follows a general Brownian motion with a drift rate of 0.1 per month, and a volatility rate of 0.16 per month. The initial cash position is 2.0 That is, the cash position at time LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1GLDYlUSJ0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0Y4USdub3JtYWxGJ0YrRjo= follows the SDE dX = 0.1 dt + 0.16 \134; dW \134\134 X(0) = 2.0 Read the problem and the SDE carefully! (a) What are the probability distributions of the cash position after 1 month, 6 months, and 1 year? (b) What are the probabilities of a negative cash position at the end of 6 months and one year?
<Text-field style="Heading 2" layout="Heading 2">(a)</Text-field> The cash position at 1 month, 6 months, and 1 year are respectively X(1) ~ N(2.1, (0.16)^2) X(6) ~ N(2.6, 6*(0.16)^2) X(12) ~ N(3.2, 12*(0.16)^2)
<Text-field style="Heading 2" layout="Heading 2">(b)</Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJStleGVjdXRhYmxlR0Y9Rjk= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRK1N0YXRpc3RpY3NGJ0YvRjIvRjNRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiOkYnRj0vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkUvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGVA== print(); # input placeholder 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder SSNfUkc2JSUqcHJvdGVjdGVkR0kvUmFuZG9tVmFyaWFibGVHNiJJOV9Qcm9iYWJpbGl0eURpc3RyaWJ1dGlvbkdGJg== \134Pr[ X(1) < 0 ] &=& \134Pr[ (Z-2.1)/0.16 < -2.1/0.16] = 1.1E-39; 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder JCIzYDw+Jkd6OFI9IiEjYw== \134Pr[ X(6) < 0 ] &=& \134Pr[ (Z-2.6)/0.96 < -2.6/0.96] = 0.00338; 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LCYjIiIiIiIjRiQqJkYjRiQtSSRlcmZHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywkKigkIittbTthOCEiKkYkKSIiJ0YjRiQpRiVGI0YkRiRGJCEiIg== LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi5RJyZyYXJyO0YnLyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZHLUkmbXRleHRHRiQ2JVEtYXR+MjB+ZGlnaXRzRidGL0YyRkM= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbWlHRiQ2KVEmZXZhbGZGJy8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYnLyUlYm9sZEdRJmZhbHNlRicvJSdpdGFsaWNHUSV0cnVlRicvJStmb3JlZ3JvdW5kR1EoWzAsMCwwXUYnLyUrZXhlY3V0YWJsZUdGNC8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYxUSJbRidGL0YyRjhGOy9GPlEnbm9ybWFsRicvJSZmZW5jZUdGNy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y3LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjE2NjY2NjdlbUYnLyUncnNwYWNlR0ZWLUkjbW5HRiQ2KFEjMjBGJ0YvRjJGOEY7RkQtRkE2MVEiXUYnRi9GMkY4RjtGREZGRkhGSkZMRk5GUEZSRlQvRlhRLDAuMTExMTExMWVtRictRkE2MVEiKEYnRi9GMkY4RjtGREZGRkhGSkZMRk5GUEZSRlRGVy1GQTYxUSJ+RidGL0YyRjhGO0ZEL0ZHRjRGSC9GS0Y0RkxGTkZQRlIvRlVRJjAuMGVtRicvRlhGZW8tSShtYWN0aW9uR0YkNiUtSSptdmVyYmF0aW1HRiQ2I1FdciwmIyIiIiIiIyIiIiomIyIiIiIiIyIiIi1JJGVyZkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCQqKCQiK21tO2E4ISIqIiIiKSIiJyMiIiIiIiMiIiIpIiIjIyIiIiIiIyIiIiIiIiIiIiEiIkYnLyUrYWN0aW9udHlwZUdROW1hcGxlc29mdC5jb206bGFiZWwoTDgzKUYnLyUldmlld0dRJmxhYmVsRidGX28tRkE2MVEiKUYnRi9GMkY4RjtGREZGRkhGSkZMRk5GUEZSRlRGVw== JCIrY1o7TDshIz8= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= \134Pr[ X(12) < 0 ] &=& \134Pr[ (Z-3.2)/1.92 < -3.2/1.92] = 0.04779; 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LCYjIiIiIiIjRiQqJkYjRiQtSSRlcmZHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywkKigkIittbW1tOyEiKkYkKSIiJEYjRiQpRiVGI0YkRiRGJCEiIg== LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi5RJyZyYXJyO0YnLyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZHLUkmbXRleHRHRiQ2JVEtYXR+MjB+ZGlnaXRzRidGL0YyRkM= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbWlHRiQ2KVEmZXZhbGZGJy8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYnLyUlYm9sZEdRJmZhbHNlRicvJSdpdGFsaWNHUSV0cnVlRicvJStmb3JlZ3JvdW5kR1EoWzAsMCwwXUYnLyUrZXhlY3V0YWJsZUdGNC8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYxUSJbRidGL0YyRjhGOy9GPlEnbm9ybWFsRicvJSZmZW5jZUdGNy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y3LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjE2NjY2NjdlbUYnLyUncnNwYWNlR0ZWLUkjbW5HRiQ2KFEjMjBGJ0YvRjJGOEY7RkQtRkE2MVEiXUYnRi9GMkY4RjtGREZGRkhGSkZMRk5GUEZSRlQvRlhRLDAuMTExMTExMWVtRictRkE2MVEiKEYnRi9GMkY4RjtGREZGRkhGSkZMRk5GUEZSRlRGVy1GQTYxUSJ+RidGL0YyRjhGO0ZEL0ZHRjRGSC9GS0Y0RkxGTkZQRlIvRlVRJjAuMGVtRicvRlhGZW8tSShtYWN0aW9uR0YkNiUtSSptdmVyYmF0aW1HRiQ2I1FdciwmIyIiIiIiIyIiIiomIyIiIiIiIyIiIi1JJGVyZkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCQqKCQiK21tbW07ISIqIiIiKSIiJCMiIiIiIiMiIiIpIiIjIyIiIiIiIyIiIiIiIiIiIiEiIkYnLyUrYWN0aW9udHlwZUdROW1hcGxlc29mdC5jb206bGFiZWwoTDg3KUYnLyUldmlld0dRJmxhYmVsRidGX28tRkE2MVEiKUYnRi9GMkY4RjtGREZGRkhGSkZMRk5GUEZSRlRGVw== JCItP0EkPT8pUSEjPw== JSFH
JSFH