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Partial Differential Equations and Probability Density Functions

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Please see the attached file for the fully formatted problems.
1) Suppose that S is a random variable that is defined on [0,∞) and whose probability density function is:

(see the attached file)

a and b being positive numbers. Show:

where

(see the attached file)

2) We know that the solution of the final value problem

(see the attached file)

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Partial differential equations and PDFs are investigated.

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