Compute expected value of the event

Question

Let $S = \{ 1, 2, 3, 4, 5, 6, 7, 8 \}$ be a sample space with $P(x) = k^{2}x$ where $x$ is a member of $S$, and $k$ is a positive constant. Compute $\mathbb{E}(S)$. Round your answer to the nearest hundredths.

I have tried solving this problem by substituting the set values and using probability formulas, but everything has been incorrect.

Answer 1

You've been given a really poorly stated question.

First of all, $S$ is defined as a set and then you are asked for $\textbf{E}(S)$, which does not make sense. Instead, I would interpret the question as:

Let $X$ be a random element selected from the set $S$, with $P(X=x)=P(x)=k^2x$, where $k$ is a positive constant. Find $\textbf{E}(X)$.

Now as Henry states, you need to solve for the value of $k$. The fact that you will need to use here is the sum of all the $P(x)$'s must be 1.

Next, you can simply apply the definition of expected value: $\textbf{E}(X)=\sum_{x} x P(x)$, where the sum is over all valid values of $x$ (i.e., all $x \in S$).