expected value rule of Uniform random variable

Question

Let be a uniform random variable on the range $\{−1,0,1,2\}$. Let $=^4$. Use the expected value rule to calculate $\mathbb{E}[Y]$.

Why isn’t the answer $1/4 \cdot(-1)^4 +1/4\cdot (0)^4+ 1/4 \cdot(1)^4 + 1/4 \cdot(2)^4$

Answer 1

The rv X is the following

$$X = \begin{cases} \frac{1}{4}, & \text{if $x=-1$} \\ \frac{1}{4}, & \text{if $x=0$} \\ \frac{1}{4}, & \text{if $x=1$} \\ \frac{1}{4}, & \text{if $x=2$} \end{cases}$$

If you transform $Y=X^4$ when $X=\pm1$ you get that $Y=1$ in both cases, so you have

$$Y = \begin{cases} \frac{1}{4}, & \text{if $y=0$} \\ \frac{1}{2}, & \text{if $y=1$} \\ \frac{1}{4}, & \text{if $y=16$} \end{cases}$$

Thus

$$\mathbb{E}[Y]=\frac{1}{2}+4=\frac{9}{2}$$

Next time, please use MathJax to format your formulas; it is easy and your message is more readeable

In any case, your answer is correct...

$$\mathbb{E}[Y]=(-1)^4\times\frac{1}{4}+(0)^4\times\frac{1}{4}+(1)^4\times\frac{1}{4}+(2)^4\times\frac{1}{4}=\frac{9}{2}$$