Using Mathematica to study basic probability

by Nasser Abbasi, september 30, 2007

Basic relations. Let f(x) be a PDF for some continuous random variable.  Then the following is true
As can be verified as follows (using Normal distribution as an example) by evaluating the above integral and see if it the same as CDF(X=a)





Now find F (a), it should be the same as above



Another important relation is probability of X being in some range. This is the same as the area under the curve of f(x) between the 2 points:

The above is found as follows



Now find F (b)-F(a), it should be the same as above




lets try to see how to find probabilty of X be in some range when X is discrete. Assume X is a discrete random variable, say a Binomial. We need to do the same as above. Now we can not use Integrate, but need to us SUM


Graphics:Example of binomial

Let find P (X < 3) in the above. Now instead of integration, we use sum, we want to add the area under the PDF from 0 to 3. But the width is 1 for each interval. So we just sum the y values.



Verify by checking the CDF at 3, it should be the same as above



To show that probability mass function adds to one. Say we have binomial distribution





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