Science:Math Exam Resources/Courses/MATH103/April 2014/Question 03 (c)

MATH103 April 2014

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Question 03 (c)

Vancouver has some of the best salmon hatcheries (i. e. salmon farms) in the world. The probability that a young salmon takes $t$ days to swim from the Capilano Salmon Hatcher to the Pacific Ocean is given by the probability density function $p(t)=e^{3-t}$ for $3\leq t\leq \infty$. [It is assumed that it takes at least $3$ days for a salmon to reach the ocean.] Assume that the lifetime of a salmon depends on how quickly it manages to reach the Pacific Ocean as a juvenile. Suppose that a salmon lives $s$ years, where $s=\ln(t+1)$, if it reach the Pacific Ocean in $t$ days. Find the probability density function $q(s)$ for $s$. Work must be shown for full marks. Simplify fully.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

Hint 1
Note the first sentence of the second paragraph. What does it say about the relationship between the probability density functions of q(s) and p(t)?

Hint 2
Use $q(s)\mathrm {d} s=p(t)\mathrm {d} t$.

Hint 3
Your final answer should be a function of $s$ (not $t$).

Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution 1

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Since $s=\ln(1+t)$, we have $\mathrm {d} s={\frac {1}{1+t}}\mathrm {d} t$, and so ${\begin{aligned}q(s)\mathrm {d} s=p(t)\mathrm {d} t&\Rightarrow {\frac {q(s)}{1+t}}\mathrm {d} t=e^{3-t}\mathrm {d} t\\&\Rightarrow q(s)=(1+t)e^{3-t}.\end{aligned}}$ But $s=\ln(1+t)$ implies $t=e^{s}-1$, so we get $q(s)=(1+t)e^{3-t}=(1+e^{s}-1)e^{3-(e^{s}-1)}=e^{s}e^{4-e^{s}}=e^{s-e^{s}+4}.$

Solution 2

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. For an alternative solution, let's use the cumulative density function ${\begin{aligned}F(t)&=\int _{3}^{t}p(x)\,dx=\int _{3}^{t}e^{3-x}\,dx\\&=\left.-e^{3-x}\right|_{3}^{t}=1-e^{3-t}\end{aligned}}$ Connecting the cdf of the juvenile to the lifetime of the salmon, $s=\ln(t+1)$, that is, $t=e^{s}-1$ we find the cdf of the lifetime of a salmon as ${\begin{aligned}G(s)&=F(e^{s}-1)\\&=1-e^{3-e^{s}+1}=1-e^{4-e^{s}}\end{aligned}}$ Finally, to find the pdf we take the derivative $q(s)=G'(s)=-e^{4-e^{s}}(-e^{s})=e^{4+s-e^{s}}$

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