How to Find $E(g(X))$ Where $g(x)=x^2+x+1$ Using a Random Variable Mass Function

Question

I am trying to figure out what $E(g(X))$ is, given that $g(x)=x^2+x+1$, and the random mass variable X has mass function:

So far, I have made the following steps:

1) Assuming E is expectancy, multiply and add $(0.5*1)+((1/3)*2)+((1/6)*3) = 5/3$ 2) Sub in $5/3$ into $g(x)$ to get $(5/3)^2+(5/3)+1$ = $5.444$

However, I am unsure that this is the correct answer to this question. Can anybody confirm if it is? Or, if not, what I should do in order to obtain the proper answer?

Answer 1

This is not right.

You should come up with the probabilities for each value of g

$ g(1) = (1)^2 + 1 + 1 = 3 $ with probability $ \frac{1}{2} $

$ g(2) = (2)^2 + 2 + 1 = 7 $ with probability $ \frac{1}{3} $

$ g(3) = (3)^2 + 3 + 1 = 13 $ with probability $ \frac{1}{6} $

Therefore the expected value is $ 3 \left(\frac{1}{2}\right) + 7 \left(\frac{1}{3}\right) + 13 \left(\frac{1}{6}\right) = 6 $