What are the mean and standard deviation of the probability density function given by #p(x)=k(x-x^2) # for # x in [0,1]#, in terms of k, with k being a constant such that the cumulative density across all x is equal to 1?

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Answer 1 · 2016-11-09T15:00:09

#E(X)=k/12#

#sd=sqrt((3k-5k^2)/60)#

the mean #E(X)# of a continuous pdf is given by

#E(X)=int_(all x)xf(x)dx#

so

#E(X)=int_0^1kx(x-x^2)dx#

#E(X)=kint_0^1(x^2-x^3)dx#

#E(X)=k[1/3x^3-1/4x^4]_cancel(0)^1#

#E(X)=k(1/3-1/4)=k/12#

#Var(X)=E(X^2)-E^2(X)#

#E(X^2)=int_(all x)x^2f(x)dx#

#E(X^2)=int_0^1x^2k(x-x^2)dx#

#E(X^2)=kint_0^1(x^3-x^4)dx#

#E(X^2)=k[1/4x^4-1/5x^5]_cancel(0)^1#

#E(X^2)=k(1/4-1/5)=k/20#

#Var(X)=E(X^2)-E^2(X)#

gives

#Var(X)=k/20-(k/12)^2#

#Var(X)=(3k-5k^2)/60#

#sd=sqrt((3k-5k^2)/60)#