121616949-math.234 - curve As usual we need to think about...

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220 Chapter 9 Applications of Integration Show that R -∞ f ( x ) dx = 1. Is f a probability density function? Justify your answer. 7. Find all the modes and the mean for the distribution in example 9.30 . 8. What is the expected value of one roll of a fair die? 9. what is the expected sum of one roll of three fair dice? 10. What is the mean of the uniform distribution on [ a, b ]? (See example 9.24 .) 11. Compute the variance and standard deviation of the exponential distribution from exam- ple 9.27 . 12. Compute the mean of the distribution from example 9.33 . 13. If you have access to appropriate software, compute the range of “good” values for the chip manufacturing process “at the 5% level.” That is, find r so that Z 10+ r 10 - r f ( x ) dx 0 . 95 and interpret this as a range of values for the number of defective chips that should cause no alarm. You might think about circumstances that would make the 5% or the 1% threshold
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Unformatted text preview: curve. As usual, we need to think about how we might approximate the length, and turn the approximation into an integral. We already know how to compute one simple arc length, that of a line segment. If the endpoints are P ( x , y ) and P 1 ( x 1 , y 1 ) then the length of the segment is the distance be-tween the points, p ( x 1-x ) 2 + ( y 1-y ) 2 , from the Pythagorean theorem, as illustrated in figure 9.19 . ....................................................................................... ( x 1 , y 1 ) ( x , y ) x 1-x y 1-y p ( x 1-x ) 2 + ( y 1-y ) 2 Figure 9.19 The length of a line segment. Now if the graph of f is “nice” (say, differentiable) it appears that we can approximate the length of a portion of the curve with line segments, and that as the number of segments...
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