Chapter 18.1, Problem 42E

Flea Beetles The mobility of an insect is an important part of its survival. Researchers have determined that the probability that a marked flea beetle, Phyllotreta cruciferae and Phyllotreta striolata, will be recaptured within a certain distance and time after release can be calculated from the probability density function

      $p\left(x,t\right)=\frac{e^{-x^2/\left(4Dt\right)}}{{\displaystyle {\int }_0^L{e}^{-u^2/\left(4Dt\right)}du}},$

 where t is the time after release (in hours), x is the distance (in meters) from the release point that recaptures occur, L is the maximum distance from the release point that recaptures can occur, and D is the diffusion coefficient. Source: Ecology Monographs.

(a)    If t = 12, L = 6, and D = 38.3, find the probability that a flea beetle will be recaptured within 3 m of the release point.

(b)    Using the same values for t, L, and D, find the probability that a flea beetle will be recaptured between 1 and 5 m of the release point.