What are the mean and standard deviation of the probability density function given by #(p(x))/k=x/(1-x^2# for # x in [2,4]#, in terms of k, with k being a constant such that the cumulative density across the range of x is equal to 1?

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Answer 1 · 2016-07-17T01:14:38

#k=1/-2.2939 #

PDF means that we need to find the expectation which becomes
#E(x) = int_2^4 p(x) times x # #dx#

and
#p(x) = k(x/(1-x^2)) #

#int_2^4 k(x/(1-x^2)) times x # #dx#

#kint_2^4 x^2/(1-x^2) dx#

#k[(ln(4+1)-ln(4-1)-2*4)/2 - (ln(2+1)-ln(2-1)-2*2)/2 ]#

#k(ln(5)/2-ln(3)-2)#

#k(-2.2939)#
for the cummulative value to be equal to 1 then

#-2.2939k=1#
then solve for k
#k=1/-2.2939 #