In Excel for a project I'm trying to model the density, distribution and survival function ($1-F(X)$) and I can't get the density to sum to one and I can't get the distribution to go to one. For the parameters I'm using $\alpha=3$ and $\theta=10$. I'm getting extremely strange results such that halfway through the distribution it levels of at $.36366$ until the end. The density function is summing to $1.15...$. After checking the formula that I input, I know it's right, but is there something I should know about modeling this in excel that maybe the program is picky about? I'm not sure if this is enough information, if not comment on what you need to know and I'll get back to you.
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1$\begingroup$ Can you explain more about what calculations you're actually doing? $\endgroup$– Glen_bCommented Oct 3, 2013 at 0:13
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$\begingroup$ The Pareto Type II distribution is continuous, but you are summing... you should be integrating. If you can't integrate, pick a fine grid with interval, say, 0.1, and lower and upper bounds (0, 100) (I choose this because of your parameter values). Calculate the density over that grid, and sum. You should get something very close to 10 (1/0.1). If you make the grid finer, e.g., 0.01-width intervals the result will be closer to 100, relatively speaking, and so on. $\endgroup$– jbowmanCommented Oct 3, 2013 at 0:48
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$\begingroup$ This is insufficient information. If you would like specific help, please share the formula you are using. $\endgroup$– whuber ♦Commented Oct 3, 2013 at 5:03
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$\begingroup$ Sorry in the light of the moment I suppose I wasn't exactly sure how to word correctly what I needed help with. I have a solution now and I voted to close this since the question really didn't make too much sense. $\endgroup$– KyleCommented Oct 3, 2013 at 5:23
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1$\begingroup$ @Kyle it would be better if you could find a way to edit your question that reflects what you didn't understand more clearly. The point of answering isn't just to resolve your particular issue, it's to solve the problems of future readers with similar issues. If you can edit to ask the kind of question that you would have asked if you'd known better how to put it, you can then supply an answer that would have been useful to the you that asked the original Q, and which may help the next person. $\endgroup$– Glen_bCommented Oct 3, 2013 at 5:29
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Just to be sure, in order to "mimick" integration cell-by-cell in Excel, you need to go at it in true Riemannian style. Namely, pick an equally spaced fine grid as @jbowman says, $\{x_1,...,x_n\}$ with the minimum and maximum values small and large enough so as not to leave out any serious probability mass (Pareto has of course a finite minimum value in its support). Call the step, $x_{i} -x_{i-1}=h$. Calculate the density values $f_X(x_i)$. Then approximate probability mass at each interval by
$$P\Big(X\in [x_{i-1},x_i]\Big) \approx h \cdot \frac {f_X(x_{i-1}) + f_X(x_i)}{2} $$
Then
$$P\Big(X\in [x_1,x_n]\Big) \approx \sum_{i=2}^{n}h \cdot \frac {f_X(x_{i-1}) + f_X(x_i)}{2} $$ $$= \frac {h}{2}\Big(f_X(x_{1}) + f_X(x_n)\Big) + h \sum_{i=2}^{n-1}f_X(x_{i}) $$
Is this what you are calculating?
answered Oct 3, 2013 at 1:47