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Homework Set 1

Microsoft Excel has a number of built-in statistical distribution functions (note that these are not the same functions used by @RISK for Monte Carlo sampling). Find the distribution function for the log-normal CDF.

Use the log-normal CDF to generate the graphs shown here. To generate the PDF curves, remember the relationship between the PDF and CDF, i.e. use a backwards difference approximation to the first derivative to generate the PDF curves.

Since the mode is the most likely point on the PDF, approximately what is the mode for the log-normal distribution when m = 1 and s = 1?

Since the median is the point at which cumulative probability is equal to 50 % (i.e. outcomes lower and higher than the median are equally likely), approximately what is the median for the log-normal distribution when m = 1 and s = 2?

Imagine that you have been carrying out pilot plant studies of the conversion of a reactant in a reactor. You have found that the conversion is most likely to be 0.98, but that 10 % of the time, it falls below 0.90 because of poor catalyst quality. Pick an appropriate statistical distribution and model the reactor conversion. Graph your results in Microsoft Excel.

Note that I have specified two degrees of freedom for this problem (if this is not obvious, think about it a little more). A probability distribution that requires only one parameter (i.e. has only one degree of freedom), such as the chi-square distribution, is not a good choice here.

Hint 1: Remember, you can put the distribution function in an equation to fit your data. Technically speaking, this procedure is called scaling and translating the distribution. For instance,

In this case, a is the translation constant and b is the scaling parameter.

Hint 2: You might also find the inverse distribution function useful for generating values of the random variable given a cumulative probability of an outcome.

Updated: Saturday, March 19, 2011