Discrete Random Variable?

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).

Discrete Random Variable?
To have a proper distribution, we should have


P(X=1)¨P(X=2)+...+P(X=n) = 1


so


1/c + 2/c + ...+ n/c should be = 1


or


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





The formula of Gauss gives 1+2+...+n = n(n+1)/2


hence


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


or


c = n(n+1)/2





E(X) = sum (j = 1 to n) jP(X=j) =.sum(j = 1 to n) j²/c


and now use 1²+2²+3²+.....+n²=n*(n+1)*(2n+1)/6





E(X²) = sum(j = 1 to n) j² P(X=j) = sum(j=1 to n) j³/c


and now use 1³ + 2³ + 3³ + ... + n³ = ( 1 + 2 + 3 + ... + n )²





Var(X) = E(X²) - E²(X)





E(1/X)= sum (j=1 to n) (1/j) P(X=j) = sum(j = 1 to n) 1/c = n/c

flower show