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

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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 ${\displaystyle t}$ days to swim from the Capilano Salmon Hatcher to the Pacific Ocean is given by the probability density function ${\displaystyle p(t)=e^{3-t}}$ for ${\displaystyle 3\leq t\leq \infty }$. [It is assumed that it takes at least ${\displaystyle 3}$ 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 ${\displaystyle s}$ years, where ${\displaystyle s=\ln(t+1)}$, if it reach the Pacific Ocean in ${\displaystyle t}$ days. Find the probability density function ${\displaystyle q(s)}$ for ${\displaystyle s}$.

Work must be shown for full marks. Simplify fully.

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