8 03 at ESG Supplemental Notes 8 044 Spring 2006 More on Problem 4 PS3 The probablity density has been found to be 4 2 3 2 2 vx2 2 vx exp 2 p vx 2 some of us prefer the above form for reasons that might become clear later The needed improper definite integrals are of the form 0 2 u2 u e du 4 0 3 u2 u e 1 du 2 0 4 u2 u e 3 du 8 these integrals have been treated extensively elsewhere and will not be rederived We have then using basic calculus techinques to account for the needed factors of 2 2 vx2 4 2 3 2 3 2 vx exp 2 dvx vx 2 0 2 2 2 vx2 4 2 3 2 4 vx 2 vx exp 2 dvx 2 0 4 2 3 2 2 2 1 2 2 2 4 2 3 2 2 5 2 3 2 2 8 3 2 From this we see that Var vx 3 8 2 0 4535 2 stand dev vx 0 6734 2 in this case the parameter is not the standard deviation of the probability distrtibution A plot of p vx as a function of vx is on the following page 1 What can clearly be seen is the maximum of p vx is attained at the most likely speed easily calculated as 2 slightly less than the mean vx 8 1 596 It is also seen that the range from the mean the standard deviation from 0 92 to 2 27 is a reasonable estimate of the spread of the probability distribution The above plot was generated using Excel the default spreadsheet program for Windows Use of a spreadsheet allows numerical calculation of the mean and standard deviation Excel is maybe not the best program for plotting but it s fairly common 2
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