The errorfunction defined by 2 C x erf(x) / e~'' dt -Jk JO gives the probability that

Chapter 1, Problem 24

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The errorfunction defined by 2 C x erf(x) / e~'' dt -Jk JO gives the probability that any one of a series oftrials will lie within x units ofthe mean, assuming that the trials have a normal distribution with mean 0 and standard deviation n/2/2. This integral cannot be evaluated in terms of elementary functions, so an approximating technique must be used. a. Integrate the Maclaurin series for e~x ' to show that erf(x) * (2k+\)k\ b. The error function can also be expressed in the form 2 2 k x 2k+] erf(x) e > . V* 1 3-5---(2 +1) Verify thatthe two series agree for A = 1, 2, 3, and 4. [Hint: Use the Maclaurin series for e~x '.] c. Use the series in part (a) to approximate erf(l) to within ID-7 . d. Use the same number ofterms as in part (c) to approximate erf(l) with the series in part (b). e. Explain why difficulties occur using the series in part (b) to approximate erf(x).