Let X be a random variable with the following p.m.f. f(x) = cx(4 − x), x = 1, 2, 3 Let $X$ be a random variable with the following p.m.f. $f(x) = cx(4 − x), x = 1, 2, 3$
and zero, otherwise.

(i) Find c

(ii) Find $E(X)$ and $Var(X).$

(iii) Find the moment generating function.

(iv) Draw a graph of the probability histogram of the pmf of $X$

I was able to find the value for c by doing $1=c(1(4-1) + 2(4-2) + 3(4-3))$ which gives $c = 1/10.$
i) Yes, $c=1/10$, not $c=1/11$ as originally stated.

ii) from iii)

iii) $M_{X}(t)=\mathbb{E}[e^{tX}]=e^{t}*3/10+e^{2t}*4/10+e^{3t}*3/10$

$M_{X}'(0)=(3+8+9)/10=2$ and $M_{X}''(0)=(3+16+27)/10=46/10$
So what you do is we know $E(x)=x(f(x))$
And V$ar(x)=x^2(f(x))-(E(x))^2$

All you do is plug the numbers in and its very easy from then on