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Unformatted text preview: 2.1. SIMULATION OF CONTINUOUS PROBABILITIES 47 0.5 1 1.5 2 0.2 0.4 0.6 0.8 1 Figure 2.7: Sum of two random numbers. [ a,b ] approximates the probability that a ≤ X ≤ b . But the sum of the areas of these bars also approximates the integral b a f ( x ) dx . This suggests that for an experiment with a continuum of possible outcomes, if we find a function with the above property, then we will be able to use it to calculate probabilities. In the next section, we will show how to determine the function f ( x ). Example 2.5 Suppose that we choose 100 random numbers in [0 , 1], and let X represent their sum. How is X distributed? We have carried out this experiment 10000 times; the results are shown in Figure 2.8. It is not so clear what function fits the bars in this case. It turns out that the type of function which does the job is called a normal density function. This type of function is sometimes referred to as a “bell-shaped” curve. It is among the most important functions in the subjectas a “bell-shaped” curve....
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- Spring '09