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This density is referred to as the uniform density over the range [ 05, 05] 1 For N = 2, in MATLAB use the rand function to produce instances of xi for i = 1, , N Then compute xN according to eqn (420) with A = 1 Repeat this experiment M = 10, 000 times to produce N M realizations of xN (a) Graph the histogram of the M realizations of xN with the axis limits set to 10 using about 20 bins (b) Compute the mean and the standard deviation of the M realizations of xN (c) On the same axes as the histogram, graph the Gaussian density with and determined in the previous step 2 Repeat the above process for N = 1, 3, and 10 Compare the results with the statement of the central limit theorem While this problem is not a proof of the central limit theorem, it visually shows that the density of the summed random variables rapidly converges toward a Gaussian density as N increases An analytic demonstration using circumstances related to this problem is contained in Section 113 in [31] Exercise 42 Compute the mean, variance, and second moment of the random variable x with the uniform density over the interval (0, 1): px (X) = 1, 0, for X (0, 1) otherwise

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AAII is a nonprofit educational organization founded in 1978 by James Cloonan Members are typically nearing or in retirement and have a relatively high net worth One of the organization s goals is to educate individual investors to manage their own portfolios In 1987, AAII started polling individual members each week about the stock market Before the Internet, random AAII members were polled by postcards; since 2000, all members have to do is vote online It wasn t long before the financial world discovered that the poll results could be used as a contrarian indicator In other words, if members are feeling excessively bullish or bearish, traders could do well by doing the exact opposite

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Exercise 43 For the uniform density de ned in Exercise 42, nd 1 P {x < 0} 2 P {x < 025} 3 P {x < 025 or x > 050} 4 P {x < 025 or x < 050} 5 P {x < a} for a R Exercise 44 Find the density for y where y = x3 and x N (0, 2 ) Exercise 45 Find the density for y where y = x2 and px (X) = X exp( X), 0, for x > 0 otherwise

The density for x is a special case of the Gamma or Erlang density Exercise 46 If z = x2 where x N (0, 2 ), use the method of Example 45 to show that pz (Z) =

There is nothing more fascinating than getting inside the heads of individual investors After all, the market is driven by twin emotions of fear and greed, as pointed out by traders such as Jesse Livermore or investors such as Warren Buffett Therefore, it is not surprising that one of the most watched is the weekly AAII Sentiment Survey The results are published on the AAII Web site or in financial periodicals such as Barron s and give insights into the mind of the little guy At times, the survey can be uncannily accurate that is, it can be if you do the opposite of what the members are feeling The survey gets interesting when we see levels of excess, says Charles Rotblut, vice president and AAII Journal editor It

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