Question #d032c

1 Answer
MattyMatty
Dec 8, 2017

#1/theta#

Explanation:

#f(x) = theta*e^(-theta*x)#
First moment is the mean or E[x] :
#E[x] = int x*theta*e^(-theta*x) dx#
#E[x] = theta * int x*e^(-theta*x) dx#
Partial integration yields
#E[x] = -x*e^(-theta*x) - int e^(-theta*x) dx#
#E[x] = -x*e^(-theta*x) - e^(-theta*x)/theta#
Evaluation between 0 and #oo# yields
#E[x] = - 0 - 0 + 0 + 1/theta = 1/theta#

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