Science:Math Exam Resources/Courses/MATH103/April 2017/Question 09 (c)

MATH103 April 2017

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Question 09 (c)
The probability density function (pdf ) for the mortality of a jellyfish (Turritopsis dohrnii), $p(x)$, at age $x$ is given by $p(x)={\frac {2}{\pi }}\cdot {\frac {1}{1+x^{2}}},$ for $0\leq x<\infty$. (c) Find the mean mortality, ${\bar {x}}$. (Setup and evaluate the improper integral for the mean mortality.)

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If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Recall the definition of mean value, and that evaluating an improper integral requires taking a limit, where the limit is applied to a proper integral.

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. The mean mortality is ${\bar {x}}=\int _{0}^{\infty }tp(t)dt={\frac {1}{\pi }}\int _{0}^{\infty }{\frac {2t}{1+t^{2}}}dt.$ The right-most integral equals $\lim _{r\to \infty }{\frac {1}{\pi }}\int _{0}^{r}{\frac {2t}{1+t^{2}}}dt.$ And this proper integral can be evaluated by using the substitution $u=1+t^{2}$, $\int {\frac {2t}{1+t^{2}}}dt=\int {\frac {du}{u}}=\ln |u|=\ln(1+t^{2})$ $\int _{0}^{r}{\frac {2t}{1+t^{2}}}dt=\ln(1+r^{2})-\ln(1+0^{2})=\ln(1+r^{2})$ where the last equality follows from the fact that $\ln(1)=0$. It follows that ${\bar {x}}={\frac {1}{\pi }}\lim _{r\to \infty }\ln(1+r^{2})=\infty$ Hence, $\color {blue}{\bar {x}}=\infty$