Need help A detective is interested on studying the traffic accidents in a town. The time between accidents in aday in this town is given by T, with density function f(t) = et for t≥ 0. Calculate the expected valueof the random variable g(T) = e-T + 1.

The probability density function (pdf) of T is f(t) = Expected value of g(T)= isE(g(T)) =…

Answered: A detective is interested on studying…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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A detective is interested on studying the traffic accidents in a town. The time between accidents in a
day in this town is given by T, with density function f(t) = et for t≥ 0. Calculate the expected value
of the random variable g(T) = e-T + 1.

Expert Solution

Step 1: Expected value is obtained.

The probability density function (pdf) of T is

f(t) = $e to the power of negative t end exponent comma space t greater or equal than 0$

Expected value of g(T)= $e to the power of negative T end exponent plus 1$ is

E(g(T)) = $integral subscript 0 superscript infinity left parenthesis e to the power of negative t end exponent plus 1 right parenthesis space e to the power of negative t end exponent space d t space equals space integral subscript 0 superscript infinity e to the power of negative 2 t end exponent d t space plus integral subscript 0 superscript infinity e to the power of negative t end exponent d t$

= $1 half plus 1$ =1.5

noting that $integral subscript 0 superscript infinity x to the power of a minus 1 end exponent e to the power of negative b x end exponent d x space equals space fraction numerator a factorial over denominator b to the power of a end fraction$