If random variable X has a probability density function of f(x)=1/x on the interval[e,e^2], what is the standard deviation of X?

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Answer 1 · 2016-07-22T14:43:06

#3.3361599515537#

if the PDF of #x# is #f(x)=1/x# on the interval of #[e,e^2]# then #int_e^(e^2) f(x) dx= 1#

The expected standard deviation is given by

#E[sigma]=sqrt(int_e^(e^2) (x-mu)^2 f(x)dx )#

#=sqrt(int_e^(e^2) (x^2-2xmu+mu^2)/xdx)#

the integral being
#x^2/2 -2mux+mu^2ln(x) #

and
#E[sigma] =sqrt( (e^4-e^2)/2 + mu^2 + 2mu - 2mu^2)#

now we need to solve for #mu# and we shall have our final answer

#E[mu]=int_e^(e^2) xf(x)dx #

#=int_e^(e^2) 1/x*x dx #

#=e^2-e = 4.6707742704716#

#E[sigma] = 3.3361599515537#