Discrete random variable??

Hi...i really need help...couldn't do it :((( here it is


A discrete random variable has range space {1, 2, . . . , n} and satisfies


P(X = j) = j/c for some number c. Find c, and then find E(X),


E(X^2), E(1/X) and Var(X).





thank you sooo soo much

Discrete random variable??
The sum of P(X=j) from j=1 to n must be equal to 1.





So 1/c(1+2+....+n) = 1





So c=1+2+...+n = n(n+1)/2





So E(X) = sum of jP(X=j)





= 2(1+4+9+16+....+n^2)/n(n+1) = 2n(n+1)(2n+1)/6n(n+1)


= (2n+1)/3





(since the sum of 1^2+2^2+...+k^2 = n(n+1)(2n+1)/6 )


E(X^2) = sum of j^2*P(X=j)





= 2(1+2^3 + 3^3 + 4^3 +....+ n^3)/n(n+1)





= 2n^2(n+1)^2/4n(n+1)


= n(n+1)/2





(since sum of 1^3 + 2^3 +.....+ n^3 = n^2(n+1)^2/4)








E(1/X) = sum of P(X=j)/j





= 2(1+1+1+...+1)/n(n+1) = 2n/n(n+1) = 2/(n+1)








Var(X) = E(X^2) - [E(X)]^2 = n(n+1)/2 - (2n+1)^2/9