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## Probability Distributions

Imagine a class with 100 students and at the end of the semester they all took the final exam. After the teacher graded the exam he noticed that a lot of students scored between 70 and 75 percent. A slightly smaller group of students must have studied a little harder as they got between 75 and 90 percent, and a similar sized group probably spent more time at the local pub because they scored between 55 and 70 percent. Even rarer were the groups that aced or totally bombed the test.

If the teacher wanted to visualize how the grades on his final exam were distributed, he could draw out a histogram (see Figure 3.2.1)that would show the number of exams with scores that fell into different ranges. If he then scaled each column in the histogram by dividing by the total number of exams (in this case, 100), the histogram would also give the teacher a rough estimate of the probability of picking an exam at random with a score between 70 and 75. If the teacher wanted to know what the probability the exam score would be between 70 and 90, he could simply add the columns that represented that range together to give himself a general idea.

If the teacher fit a curve to the scaled version of the histogram, so that the total area under the curve was equal to 1, he could use integration to estimate the probability of an exam having a score between any two points. The smaller the area between two points, the lower the probability that an exam will have that score. The larger the area between two points, the greater the probability of randomly selecting an exam in that range. These probabilities are confirmed by the original histogram.

At this point, you may be wondering why the teacher would want to use a fitted curve instead of his original histogram to determine probabilities. The reason for this is that it is easier to compare curves, and thus, use them to answer questions. If the teacher fit curves to several year's worth of histograms, he could use them to determine if having the study session two days before the exam helped or not.    Next: Comparing Two Samples: Classifying Up: Formal Questions of a Previous: Formal Questions of a   Index

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Frank Starmer 2004-05-19
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