CUMULATIVE DISTRIBUTION FUNCTIONS in .NET

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CUMULATIVE DISTRIBUTION FUNCTIONS
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An alternative method to describe the distribution of a discrete random variable can also be used for continuous random variables.
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De nition The cumulative distribution function of a continuous random variable X is
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Extending the de nition of f(x) to the entire real line enables us to de ne the cumulative distribution function for all real numbers. The following example illustrates the de nition. EXAMPLE 4-3 For the copper current measurement in Example 4-1, the cumulative distribution function of the random variable X consists of three expressions. If x 0, f 1x2 0. Therefore, F1x2 0, for x 0
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4-3 CUMULATIVE DISTRIBUTION FUNCTIONS
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The plot of F(x) is shown in Fig. 4-6. Notice that in the de nition of F(x) any can be changed to and vice versa. That is, F(x) can be de ned as either 0.05x or 0 at the end-point x 0, and F(x) can be de ned as either 0.05x or 1 at the end-point x 20. In other words, F(x) is a continuous function. For a discrete random variable, F(x) is not a continuous function. Sometimes, a continuous random variable is de ned as one that has a continuous cumulative distribution function. EXAMPLE 4-4 For the drilling operation in Example 4-2, F(x) consists of two expressions. F1x2 and for 12.5 x
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0 for
F1x2
20e 1 e
201u 12.52
201x 12.52
Therefore, F1x2 e 0 1 x x 12.5
201x 12.52
Figure 4-7 displays a graph of F(x).
F(x) 1 F(x) 1
Figure 4-6 Cumulative distribution function for Example 4-3.
Figure 4-7 Cumulative distribution function for Example 4-4.
CHAPTER 4 CONTINUOUS RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS
The probability density function of a continuous random variable can be determined from the cumulative distribution function by differentiating. Recall that the fundamental theorem of calculus states that
d dx Then, given F(x)
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f 1x2
f 1x2 as long as the derivative exists. EXAMPLE 4-5
dF1x2 dx
The time until a chemical reaction is complete (in milliseconds) is approximated by the cumulative distribution function F1x2 e 0 1 x x 0
0.01x
Determine the probability density function of X. What proportion of reactions is complete within 200 milliseconds Using the result that the probability density function is the derivative of the F(x), we obtain f 1x2 e 0 0.01e x x 0
0.01x
The probability that a reaction completes within 200 milliseconds is P1X EXERCISES FOR SECTION 4-3
4-11. Suppose the cumulative distribution function of the random variable X is 0 0.2x 0 1 5 x x x 0 5 Determine the following: 1.52 (a) P1X 1.82 (b) P1X 22 (d) P1 1 X 12 (c) P1X 4-13. Determine the cumulative distribution function for the distribution in Exercise 4-1. 4-14. Determine the cumulative distribution function for the distribution in Exercise 4-3. 4-15. Determine the cumulative distribution function for the distribution in Exercise 4-4. 4-16. Determine the cumulative distribution function for the distribution in Exercise 4-6. Use the cumulative distribution function to determine the probability that a component lasts more than 3000 hours before failure. 4-17. Determine the cumulative distribution function for the distribution in Exercise 4-8. Use the cumulative distribution function to determine the probability that a length exceeds 75 millimeters.
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Determine the following: (a) P1X 2.82 (b) P1X 1.52 (c) P1X 22 (d) P1X 62 4-12. Suppose the cumulative distribution function of the random variable X is 0 0.25x 1 x x x 2 2
F1x2
4-4 MEAN AND VARIANCE OF A CONTINUOUS RANDOM VARIABLE
Determine the probability density function for each of the following cumulative distribution functions. 4-18. 4-19. F1x2 1 e
4-20. 0 0.25x 0.5 0.5x 0.25 1 x x x x 2 1 1.5
0 F1x2
F1x2
0 0.2x 0.04x 1
0 0.64 4 9
x x x x
0 4 9
2 1 1.5
4-21. The gap width is an important property of a magnetic recording head. In coded units, if the width is a continuous random variable over the range from 0 x 2 with f(x) 0.5x, determine the cumulative distribution function of the gap width.