Great Expectations, Almost Sure FailureAn investor, Rosie, is offered what appears to be a great opportunity to invest in Risky Resources. Every week the value of her investment will either increase by 40% (with probability 0.55) or decrease by 40% (with probability 0.45), the results in different weeks being statistically independent, and this will continue indefinitely. (In the real world, Rosie should be skeptical and ask how we know that this is the case. But this is an example for a Mathematics course, where the author of a problem has unlimited power to set up the conditions of that problem, and the student must have complete faith in those conditions)Having studied Probability, Rosie can calculate the expected return on this investment. If the value of her initial investment is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdGPUY+, then after one week the value LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+ will be either LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW5HRiQ2JFEkMS40RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEifkYnRi8vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjgvJSlzdHJldGNoeUdGOC8lKnN5bW1ldHJpY0dGOC8lKGxhcmdlb3BHRjgvJS5tb3ZhYmxlbGltaXRzR0Y4LyUnYWNjZW50R0Y4LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGRy1JJW1zdWJHRiQ2JS1JI21pR0YkNiVRIldGJy8lJ2l0YWxpY0dRJXRydWVGJy9GMFEnaXRhbGljRictRiM2JS1GLDYkUSIwRidGL0ZRRlQvJS9zdWJzY3JpcHRzaGlmdEdGWkYv with probability 0.55 or LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW5HRiQ2JFEkMC42RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEifkYnRi8vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjgvJSlzdHJldGNoeUdGOC8lKnN5bW1ldHJpY0dGOC8lKGxhcmdlb3BHRjgvJS5tb3ZhYmxlbGltaXRzR0Y4LyUnYWNjZW50R0Y4LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGRy1JJW1zdWJHRiQ2JS1JI21pR0YkNiVRIldGJy8lJ2l0YWxpY0dRJXRydWVGJy9GMFEnaXRhbGljRictRiM2JS1GLDYkUSIwRidGL0ZRRlQvJS9zdWJzY3JpcHRzaGlmdEdGWkYv with probability 0.45, so the expected value is My8tSSJFRzYiNiMmSSJXR0YmNiMiIiIsJi1JIipHJSpwcm90ZWN0ZWRHNiQkIiNiISIjLCQmRik2IyIiISQiIzkhIiJGKy1GLjYkJCIjWEYzLCRGNSQiIidGOkYrL0YsLCRGNSQiJC8iRjM=This will continue in future weeks. I'll define a sequence of independent random variables LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiWEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImlGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn so that 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. Thus after n weeks her investment's value is 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. Since each 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 and the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiWEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImlGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn are independent, 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
Four percent expected growth per week! In one 52-week year, on the average, her initial investment has been multiplied by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1cEdGJDYlLUkjbW5HRiQ2JFElMS4wNEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYlLUYvNiRRIzUyRidGMi8lJ2l0YWxpY0dRJXRydWVGJy9GM1EnaXRhbGljRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2LVEifkYnRjIvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkgvJSlzdHJldGNoeUdGSC8lKnN5bW1ldHJpY0dGSC8lKGxhcmdlb3BHRkgvJS5tb3ZhYmxlbGltaXRzR0ZILyUnYWNjZW50R0ZILyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGV0Yy which is almost 7.7. Move over, Bill Gates and Warren Buffett, here comes Rosie!Unfortunately, this picture is not borne out by reality.
Pressing the "Go" button below will simulate this process several times, starting with the same initial investment, and plot the results (each simulation in a different colour). The number of simulations and the duration in weeks can be specified. The plot can be viewed with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= axis in either linear mode (where each unit of height corresponds to a fixed amount of money) or logarithmic mode (which is good for seeing what happens when the value becomes very large or very small).
use DocumentTools in
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use:
# Do( %component_name );
# and
# Do( %component_name = value );
# to set and get properties of the component.
# You can also use arbitrary expressions
# involving components, e.g.:
# Do( %target = %input1 + 2*%input2 );
# Note the %-prefix to each component name.
# See ?CustomizingComponents for more information.
Do(%Plot0 = display(PL0, axis[2]=[mode=linear]));
end use;
use DocumentTools in
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use:
# Do( %component_name );
# and
# Do( %component_name = value );
# to set and get properties of the component.
# You can also use arbitrary expressions
# involving components, e.g.:
# Do( %target = %input1 + 2*%input2 );
# Note the %-prefix to each component name.
# See ?CustomizingComponents for more information.
Do(%Plot0 = display(PL0, axis[2]=[mode=log]))
end use;
use DocumentTools in
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use:
# Do( %component_name );
# and
# Do( %component_name = value );
# to set and get properties of the component.
# You can also use arbitrary expressions
# involving components, e.g.:
# Do( %target = %input1 + 2*%input2 );
# Note the %-prefix to each component name.
# See ?CustomizingComponents for more information.
PL0:= plotwalks(0.6,1.4,0.55, evalb(GetProperty(RadioButton1,value)="true"));
Do( %Plot0 = PL0)
end use;
LSUlUExPVEc2JS0lJVZJRVdHNiQ7JCIiISEiIiQiJStdISIiOyQiIiEhIiIkIiYrKyIhIiItJStBWEVTTEFCRUxTRzYkUSE2IlEhNiItJSVST09URzYnLSUpQk9VTkRTX1hHNiMkIiRTJiEiIi0lKUJPVU5EU19ZRzYjJCIkPyIhIiItJS1CT1VORFNfV0lEVEhHNiMkIiVxXyEiIi0lLkJPVU5EU19IRUlHSFRHNiMkIiU1TSEiIi0lKUNISUxEUkVORzYiAlthough Rosie's investment may do very well for a while (sometimes spectacularly so), eventually its value will decline towards 0. What is going on? Was there something wrong with the expected value calculation?
<Text-field style="Heading 1" layout="Heading 1">Expected versus Almost Sure </Text-field>What's actually happening here is that there is a disconnect between the expected value and the almost sure behaviour. In effect, Rosie is betting on enormous returns which have extremely low probabilities. To get a more realistic picture of the situation, we need to see what will happen with high probability, in fact with probability 1, in the limit as time goes to \342\210\236.Now the usual limit theorems of probability that talk about "probability 1" deal with sums of independent random variables. What we have here is a product rather than a sum. Fortunately, we can convert a product to a sum using the logarithm function. Thus I will define 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 and 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 so that LyZJIldHNiI2I0kibkdGJSomJkYkNiMiIiEiIiItSSRleHBHRiU2IyZJIlNHRiVGJkYs. We can then apply the Law of Large Numbers and Central Limit Theorem to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRIm5GJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn. We can start by calculating LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiRUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUklbXN1YkdGJDYlLUYsNiVRIllGJ0YvRjItRiM2JS1GLDYlUSJpRidGL0YyRi9GMi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0YzUSdub3JtYWxGJ0ZIRkg= .
My8tSSJFRzYiNiMmSSJZR0YmNiNJImlHRiYsJi1JJGxvZ0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJjYjJCIjOSEiIiQiI2IhIiMtRi42IyQiIidGNSQiI1hGOC9GLCQhKjEhPSJbJSEjNQ==This is a negative number. According to the Law of Large Numbers, with probability 1 we will have 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 as 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, and in particular 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, which means 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. That is, it is certain that the investment value will approach 0 as time goes on. Of course it can never get to exactly 0, but eventually it will be close enough for all practical purposes. More detail can be obtained using the Central Limit Theorem. If LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiWUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImlGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn has mean LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NTY7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy and standard deviation LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NjM7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy, for large n we can approximate LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRIm5GJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn by a normal distribution with mean LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEobiYjOTU2O0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and standard deviation LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbXNxcnRHRiQ2Iy1GIzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GOFEnbm9ybWFsRictRjE2JVEnJiM5NjM7RicvRjVRJmZhbHNlRidGOkY6. As above, 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, and it's not hard to calculate JSFH\317\203 = (log(1.4) - log(0.6)) 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictSSNtbkdGJDYkUS0wLjQyMTUyNTM2MzNGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJStmb3JlZ3JvdW5kR1EqWzAsMCwyNTVdRicvJSlyZWFkb25seUdRJXRydWVGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKjJEfk91dHB1dEYnRjM=.
We find, for example, that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFElMjAwMEYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+ has mean 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 and standard deviation 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. The probability that 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, i.e. that after 2000 weeks the investment is still worth at least as much as its initial value, is approximately the probability that a normal random variable with this mean and standard deviation is nonnegative. That's extremely small (approximately LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbW5HRiQ2JFEjMTBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2Ji1JI21vR0YkNi1RKiZ1bWludXMwO0YnRjIvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGTC1GLzYkUSI2RidGMi8lJ2l0YWxpY0dRJXRydWVGJy9GM1EnaXRhbGljRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjI=). Rosie might as well hope to win a lottery as hope to break even after 2000 weeks. Even without doing the actual calculation you can see that the result should be very small, because 0 is about 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 standard deviations above the mean. Anything more than about 3 standard deviations above the mean for a normal distribution has very low probability.In the table below, you can enter a number of weeks, an initial investment and an investment target. When you press the Go button, Maple will calculate the probability that after that many weeks Rosie's investment will be worth at least as much as the target. This is done in two ways: "Probability to achieve" is calculated using the Binomial distribution for the number of times the investment gains in the given time period, and "Normal approximation" is calculated using a normal distribution (with continuity correction).use DocumentTools in
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use:
# Do( %component_name );
# and
# Do( %component_name = value );
# to set and get properties of the component.
# You can also use arbitrary expressions
# involving components, e.g.:
# Do( %target = %input1 + 2*%input2 );
# Note the %-prefix to each component name.
# See ?CustomizingComponents for more information.
if not type(DataTable3[1,1],posint) then
error "Duration must be a positive integer, got %1", DataTable3[1,1]
elif not type(DataTable3[2,1],positive) then
error "Initial investment must be positive, got %1",DataTable3[2,1]
elif not type(DataTable3[3,1],positive) then
error "Investment target must be positive, got %1",
DataTable3[3,1]
else
Svalue:= log(DataTable3[3,1]/DataTable3[2,1]);
Bvalue:= ceil((Svalue - DataTable3[1,1]*log(0.6))/log(1.4/0.6));
Do( %MathContainer0 = Binprob(DataTable3[1,1], 0.55, Bvalue));
Do( %MathContainer1 = Norprob(DataTable3[1,1], 0.55, Bvalue-0.5));
end if;
end use;
Probability to achieveNormal approximationJSFH
<Text-field style="Heading 1" layout="Heading 1">What should Rosie do?</Text-field>There are many strategies that Rosie could use to avoid having the investment's value go almost surely to 0, at the cost of a more modest growth in expected value. One piece of good advice that one always hears is "Diversify! Don't keep all your eggs in one basket". That can work here too.Let's suppose instead of putting everything into this investment, Rosie prudently keeps a certain fraction c of her wealth in cash and puts the other LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW5HRiQ2JFEiMUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RKCZtaW51cztGJ0YvLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRkctSSNtaUdGJDYlUSJjRicvJSdpdGFsaWNHUSV0cnVlRicvRjBRJ2l0YWxpY0YnRi8= into the investment. The cash earns no interest, but it doesn't lose any value either. Each week, as the investment goes up or down, she will either sell some or buy more so that the fraction of cash remains equal to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= (I'm ignoring any transaction costs for doing this). In a week where the investment gains 40%, her total wealth is multiplied by 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, while in a week where the investment loses 40%, it is multiplied by 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. Thus Rosie's expected value is multiplied by 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. Of course, this is less than what the expected value would be with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk=, but the benefit will become apparent if we do our calculation with logarithms as before. The new \316\274 is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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. As we saw, that was negative when LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk=, but it will turn out to be positive for some values of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. Here's a 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little bit of calculus will find the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= that maximizes LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NTY7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy, thus giving the best possible return in terms of the almost sure limit of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbWZyYWNHRiQ2KC1JJW1zdWJHRiQ2JS1JI21pR0YkNiVRIlNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiM2JS1GMjYlUSJuRidGNUY4RjVGOC8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRjsvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRkgvJSliZXZlbGxlZEdRJmZhbHNlRicvRjlRJ25vcm1hbEYn. The derivative 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.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Setting that = 0 and solving gives us LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSI9RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjI3Nzc3NzhlbUYnL0ZORlNGNS1JI21uR0YkNiRRJTAuNzVGJ0Y5Rjk=. As we see in the graph, that is a maximum, and the value of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NTY7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSI9RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjI3Nzc3NzhlbUYnL0ZORlNGNS1JI21uR0YkNiRRJTAuNzVGJ0Y5Rjk= is a positive number, which turns out to be approximately LUkjbW5HNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JFEuMC4wMDUwMDgzNjY4M0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJw==.With 75% of her wealth in cash, Rosie's expected wealth is multiplied by a more modest 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 each week: only one percent expected growth per week. But more importantly, she now will have LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbWZyYWNHRiQ2KC1JJW1zdWJHRiQ2JS1JI21pR0YkNiVRIlNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiM2JS1GMjYlUSJuRidGNUY4RjVGOC8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRjsvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRkgvJSliZXZlbGxlZEdRJmZhbHNlRictRjI2I1EhRicvRjlRJ25vcm1hbEYn approaching this positive value of \316\274 almost surely. In particular, with probability 1 her investment will eventually be worth a lot of money (although it may take a rather long time). Again, the Central Limit Theorem can help quantify that: as mentioned LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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, while LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEnJiM5NjM7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIn5GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGSUYy turns out to be (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.Here is a simulation of Rosie's results with this more prudent plan. The cash fraction LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= can be adjusted to anything from 0 to 1.use DocumentTools in
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use:
# Do( %component_name );
# and
# Do( %component_name = value );
# to set and get properties of the component.
# You can also use arbitrary expressions
# involving components, e.g.:
# Do( %target = %input1 + 2*%input2 );
# Note the %-prefix to each component name.
# See ?CustomizingComponents for more information.
Do(%Plot1 = display(PL1, axis[2]=[mode=linear]));
end use;
use DocumentTools in
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use:
# Do( %component_name );
# and
# Do( %component_name = value );
# to set and get properties of the component.
# You can also use arbitrary expressions
# involving components, e.g.:
# Do( %target = %input1 + 2*%input2 );
# Note the %-prefix to each component name.
# See ?CustomizingComponents for more information.
Do(%Plot1 = display(PL1, axis[2]=[mode=log]));
end use;
use DocumentTools in
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use:
# Do( %component_name );
# and
# Do( %component_name = value );
# to set and get properties of the component.
# You can also use arbitrary expressions
# involving components, e.g.:
# Do( %target = %input1 + 2*%input2 );
# Note the %-prefix to each component name.
# See ?CustomizingComponents for more information.
PL1:= plotwalks2(0.6,1.4,0.55, evalb(GetProperty(RadioButton3,value)="true"));
Do( %Plot1 = PL1)
end use;
LSUlUExPVEc2JS0lJVZJRVdHNiQ7JCIiISEiIiQiJStdISIiOyQiIiEhIiIkIiYrKyIhIiItJStBWEVTTEFCRUxTRzYkUSE2IlEhNiItJSVST09URzYnLSUpQk9VTkRTX1hHNiMkIiRTJiEiIi0lKUJPVU5EU19ZRzYjJCIkPyIhIiItJS1CT1VORFNfV0lEVEhHNiMkIiVxXyEiIi0lLkJPVU5EU19IRUlHSFRHNiMkIiU1TSEiIi0lKUNISUxEUkVORzYiIn the table below, you can enter the number of weeks, initial investment, cash fraction and an investment target. When you press the Go button, Maple will calculate the probability that after that many weeks Rosie's investment will be worth at least as much as the target. This is done in two ways: "Probability to achieve" is calculated using the Binomial distribution for the number of times the investment gains in the given time period, and "Normal approximation" is calculated using a normal distribution (with continuity correction).
use DocumentTools in
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use:
# Do( %component_name );
# and
# Do( %component_name = value );
# to set and get properties of the component.
# You can also use arbitrary expressions
# involving components, e.g.:
# Do( %target = %input1 + 2*%input2 );
# Note the %-prefix to each component name.
# See ?CustomizingComponents for more information.
if not type(DataTable4[1,1],posint) then
error "Duration must be a positive integer, got %1", DataTable4[1,1]
elif not type(DataTable4[2,1],positive) then
error "Initial investment must be positive, got %1",DataTable4[2,1]
elif not type(DataTable4[3,1],nonnegative) or
DataTable4[3,1]>1 then
error "Cash fraction must be in [0,1], got %1",
DataTable4[3,1]
elif not type(DataTable4[4,1],positive) then
error "Investment target must be positive, got %1",
DataTable4[4,1]
else
cc:= DataTable4[3,1]; aa:= cc + (1-cc)*0.6;
bb:= cc + (1-cc)*1.4;
Svalue:= log(DataTable4[4,1]/DataTable4[2,1]);
Bvalue:= ceil((Svalue - DataTable4[1,1]*log(aa))/log(bb/aa));
Do( %MathContainer2 = Binprob(DataTable4[1,1], 0.55, Bvalue));
Do( %MathContainer3 = Norprob(DataTable4[1,1], 0.55, Bvalue-0.5));
end if;
end use;
Probability to achieveLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Normal approximationLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
JSFH
<Text-field style="Heading 1" layout="Heading 1">Suggestions for further investigation</Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Derive the formulas for the standard deviation \317\203, both in the original case and with a cash fraction c.Using the Central Limit Theorem, what will be the approximate median value of Rosie's worth after n weeks, i.e. a target value that she has approximately probability 0.5 of achieving? Do this both in the original case and with a cash fraction of 0.75, and verify this using the tools in this document. What about the first and third quartiles?Using the Central Limit Theorem, approximately how large must n be if (in the original case) 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 ? Approximately how large must LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= be if (in the case with a cash fraction of 0.75) 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 ? Refine your approximation using the tools in this document.What other strategies might Rosie use to try to avoid losing money? How might these be analysed? Try some other probability distributions, discrete or continuous, for the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiWEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImlGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn. Can you find some where a similar effect occurs: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiRUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUklbXN1YkdGJDYlLUYsNiVRIldGJ0YvRjItRiM2JS1GLDYlUSJuRidGL0YyRi9GMi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0YzUSdub3JtYWxGJ0ZIRkg= grows but LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRIm5GJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSJ+RicvRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZILyUpc3RyZXRjaHlHRkgvJSpzeW1tZXRyaWNHRkgvJShsYXJnZW9wR0ZILyUubW92YWJsZWxpbWl0c0dGSC8lJ2FjY2VudEdGSC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlctRkE2LVEoJnNyYXJyO0YnRkRGRkZJRktGTUZPRlFGU0ZVRlgtSSNtbkdGJDYkRj9GREZE almost surely as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSgmc3JhcnI7RidGOUY7Rj5GQEZCRkRGRkZIRkpGTUY1LUY2Ni1RKCZpbmZpbjtGJ0Y5RjtGPkZARkJGREZGRkhGSkZNRjk=?TTdSMApJNlJUQUJMRV9TQVZFLzEzNjU4ODM3MlgsJSlhbnl0aGluZ0c2IkYlW2dsISElISEhIyQiJCIiIiUrNSIiJiIkKyZGJQ==TTdSMApJNlJUQUJMRV9TQVZFLzEzNjU4ODUwMFgsJSlhbnl0aGluZ0c2IkYlW2dsISElISEhIyQiJCIiIiQrJiIlKzVGJ0YlTTdSMApJNlJUQUJMRV9TQVZFLzEzNjU4ODQzNlgsJSlhbnl0aGluZ0c2IkYlW2dsISElISEhIyUiJSIiIiUrNSQiI3YhIiMiIiYiJCsmRiU=TTdSMApJNlJUQUJMRV9TQVZFLzEzNjU4ODU2NFgsJSlhbnl0aGluZ0c2IkYlW2dsISElISEhIyUiJSIiIiQrJiIlKzUkIiN2ISIjRidGJQ==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=TTdSMApJNFJUQUJMRV9TQVZFLzM2NjkxODBYLCUpYW55dGhpbmdHNiJGJVtnbCEhJSEhISMkIiQiIiIlKzUiIiYiJCsmRiU=TTdSMApJNFJUQUJMRV9TQVZFLzM2NjkzMDhYLCUpYW55dGhpbmdHNiJGJVtnbCEhJSEhISMkIiQiIiIkKyYiJSs1RidGJQ==TTdSMApJNFJUQUJMRV9TQVZFLzM2NjkyNDRYLCUpYW55dGhpbmdHNiJGJVtnbCEhJSEhISMlIiUiIiIlKzUkIiN2ISIjIiImIiQrJkYlTTdSMApJNFJUQUJMRV9TQVZFLzM2NjkzNzJYLCUpYW55dGhpbmdHNiJGJVtnbCEhJSEhISMlIiUiIiIkKyYiJSs1JCIjdiEiI0YnRiU=JSFHTTdSMApJNlJUQUJMRV9TQVZFLzEzNzc3ODc2NFgsJSlhbnl0aGluZ0c2IkYlW2dsISElISEhIyQiJCIiIiUrNSIiJiIkKyZGJQ==TTdSMApJNlJUQUJMRV9TQVZFLzEzNzc3ODg5MlgsJSlhbnl0aGluZ0c2IkYlW2dsISElISEhIyQiJCIiIiQrJiIlKzVGJ0YlTTdSMApJNlJUQUJMRV9TQVZFLzEzNzc3ODgyOFgsJSlhbnl0aGluZ0c2IkYlW2dsISElISEhIyUiJSIiIiUrNSQiI3YhIiMiIiYiJCsmRiU=TTdSMApJNlJUQUJMRV9TQVZFLzEzNzc3ODk1NlgsJSlhbnl0aGluZ0c2IkYlW2dsISElISEhIyUiJSIiIiQrJiIlKzUkIiN2ISIjRidGJQ==TTdSMApJNlJUQUJMRV9TQVZFLzE0MDAwNjk4OFgsJSlhbnl0aGluZ0c2IkYlW2dsISElISEhIyQiJCIiIiUrNSIiJiIkKyZGJQ==TTdSMApJNlJUQUJMRV9TQVZFLzE0MDAwNzExNlgsJSlhbnl0aGluZ0c2IkYlW2dsISElISEhIyQiJCIiIiQrJiIlKzVGJ0YlTTdSMApJNlJUQUJMRV9TQVZFLzE0MDAwNzA1MlgsJSlhbnl0aGluZ0c2IkYlW2dsISElISEhIyUiJSIiIiUrNSQiI3YhIiMiIiYiJCsmRiU=TTdSMApJNlJUQUJMRV9TQVZFLzE0MDAwNzE4MFgsJSlhbnl0aGluZ0c2IkYlW2dsISElISEhIyUiJSIiIiQrJiIlKzUkIiN2ISIjRidGJQ==JSFHTTdSMApJNlJUQUJMRV9TQVZFLzEzNzg0NDE1MlgsJSlhbnl0aGluZ0c2IkYlW2dsISElISEhIyQiJCIiIiUrNSIiJiIkKyZGJQ==TTdSMApJNlJUQUJMRV9TQVZFLzEzNzg0NDI4MFgsJSlhbnl0aGluZ0c2IkYlW2dsISElISEhIyQiJCIiIiQrJiIlKzVGJ0YlTTdSMApJNlJUQUJMRV9TQVZFLzEzNzg0NDIxNlgsJSlhbnl0aGluZ0c2IkYlW2dsISElISEhIyUiJSIiIiUrNSQiI3YhIiMiIiYiJCsmRiU=TTdSMApJNlJUQUJMRV9TQVZFLzEzNzg0NDM0NFgsJSlhbnl0aGluZ0c2IkYlW2dsISElISEhIyUiJSIiIiQrJiIlKzUkIiN2ISIjRidGJQ==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=TTdSMApJNlJUQUJMRV9TQVZFLzEzODA4NzM5NlgsJSlhbnl0aGluZ0c2IkYlW2dsISElISEhIyQiJCIiIiUrNSIiJiIkKyZGJQ==TTdSMApJNlJUQUJMRV9TQVZFLzEzODA4NzUyNFgsJSlhbnl0aGluZ0c2IkYlW2dsISElISEhIyQiJCIiIiQrJiIlKzVGJ0YlTTdSMApJNlJUQUJMRV9TQVZFLzEzODA4NzQ2MFgsJSlhbnl0aGluZ0c2IkYlW2dsISElISEhIyUiJSIiIiUrNSQiI3YhIiMiIiYiJCsmRiU=TTdSMApJNlJUQUJMRV9TQVZFLzEzODA4NzU4OFgsJSlhbnl0aGluZ0c2IkYlW2dsISElISEhIyUiJSIiIiQrJiIlKzUkIiN2ISIjRidGJQ==JSFHTTdSMApJNlJUQUJMRV9TQVZFLzEzNzUxNjU1NlgsJSlhbnl0aGluZ0c2IkYlW2dsISElISEhIyQiJCIiIiUrNSIiJiIkKyZGJQ==TTdSMApJNlJUQUJMRV9TQVZFLzEzNzUxNjY4NFgsJSlhbnl0aGluZ0c2IkYlW2dsISElISEhIyQiJCIiIiQrJiIlKzVGJ0YlTTdSMApJNlJUQUJMRV9TQVZFLzEzNzUxNjYyMFgsJSlhbnl0aGluZ0c2IkYlW2dsISElISEhIyUiJSIiIiUrNSQiI3YhIiMiIiYiJCsmRiU=TTdSMApJNlJUQUJMRV9TQVZFLzEzNzUxNjc0OFgsJSlhbnl0aGluZ0c2IkYlW2dsISElISEhIyUiJSIiIiQrJiIlKzUkIiN2ISIjRidGJQ==