121616949-math.226 - P n = Z 13 2 7 2 f x dx Of course we...

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212 Chapter 9 Applications of Integration EXAMPLE 9.23 Consider again the two dice example; we can view it in a way more that resembles the probability density function approach. Consider a random variable X that takes on any real value with probabilities given by the probability density function in figure 9.16 . The function f consists of just the top edges of the rectangles, with vertical sides drawn for clarity; the function is zero below 1 . 5 and above 12 . 5. The area of each rectangle is the probability of rolling the sum in the middle of the bottom of the rectangle, or P ( n ) = Z n +1 / 2 n - 1 / 2 f ( x ) dx. The probability of rolling a 4, 5, or 6 is
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Unformatted text preview: P ( n ) = Z 13 / 2 7 / 2 f ( x ) dx. Of course, we could also compute probabilities that don’t make sense in the context of the dice, such as the probability that X is between 4 and 5 . 8. 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 Figure 9.16 A probability density function for two dice. As this example illustrates, we do not require that f be continuous, but we will assume that f has only finitely many discontinuities (possibly zero), and that the discontinuities are either jump discontinuities or removable (a “hole” in the function). This guarantees...
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