hw-07-8-improper-integ-EXTRA

hw-07-8-improper-integ-EXTRA - Repeat parts (i) through (v)...

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Sheet1 Page 1 Chapter 7.8 homework EXTRA CREDIT Question A: let f(x)=exp(x)/(1+exp(x))^2 This is called the "Logistic Distribution" in probability. (i) In probability, we want to know: the integral from x=-infinity to x=+infinity of f(x) dx So please calculate it. (ii) The "mean" of the distribution is the integral from x=-infinity to x=+infinity of x*f(x) dx Please calculate it. (iii) The "variance" of the distribution is the integral from x=-infinity to x=+infinity of x^2*f(x) dx Please calculate it. (iv) The "skewness" of the distribution is the integral from x=-infinity to x=+infinity of x^3*f(x) dx Please calculate it. (v) The "kurtosis" of the distribution is the integral from x=-infinity to x=+infinity of x^4*f(x) dx Please calculate it. Question B: let f(x) = 1/(1+x^2) This is called the "Cauchy Distribution" in probability.
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Unformatted text preview: Repeat parts (i) through (v) of Question A but using this new f(x). Question C: Recall from our Chapter 7.7 homework the integral: integral of sqrt(1+(b*x/a)^2 * 1/(a^2-x^2) ) dx from x=0 to x=a. Does the integral converge? If not, does it diverge to +infinity, -infinity, or just wiggle? If it does converge, you do NOT need to find its value. (you could see what Wolfram Alpha has to say about it, though) Question D: Compute: (i) the integral from x=e to 10 of 1/(x*ln(x)) dx (hint: u-sub) (ii) the integral from x=e to 10^5 of 1/(x*ln(x)) dx (iii) the integral from x=e to 10^25 of 1/(x*ln(x)) dx (iv) the integral from x=e to 10^125 of 1/(x*ln(x)) dx (by the way, 10^125 is huge! The # of atoms in the universe is somewhere around 10^80) (v) Are these results converging or diverging?...
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