the pdf is given by
#f(x)={ (kx(2-x), 0< x<2), (0, " otherwise ") :}#
(1) to find #k#
#int_(all" "x)f(x)dx=1#
#:.int_0^2kx(2-x)dx#=1
#kint_0^2(2x-x^2)dx=1#
#k[x^2-x^3/3]_cancel(0)^2=1#
#k(4-8/3)=1#
#k=1/(4/3)=3/4#
(2) to find the mean /Expected value #E(X)#
#E(X)=int_(all " "x)xf(x)dx#
substituting and simplifying. leaving the value of#" "k" "# until the end.
#E(X)=kint_0^2(2x^2-x^3)dx#
#E(X)=k[(2x^3)/3-x^4/4]_cancel(0)^2#
#E(X)=k(16/3-4)#
#E(X)=3/4xx4/3=1#
(3) variance#=E(X^2)-E^2(X)#
need to find #E(X^2)#
#E(X^2)=int_(all" "x)x^2f(x)dx#
#E(X^2)=kint_0^2(2x^3-x^4)dx#
#E(X^2)=k[x^4/2-x^5/5]_cancel(0)^2#
#E(X^2)=k(8-32/5)#
#E(X^2)=3/4xx8/5#
#E(X^2)=6/5#
#" Var"=E(X^2)-E^2(X)=6/5-1^2=1/5#
