Let X be the random variable.
We have:
[tex]P(X=2x)=\dfrac{1}{2}[/tex]
[tex]P(X=\dfrac{x}{2})=\dfrac{1}{2}[/tex]
Compute the mean like this:
[tex]2x\times \frac{1}{2}+ \frac{x}{2}\times \frac{1}{2}= \frac{5}{4}x [/tex]
It is the expected return.
First compute the following: [tex]E(X)^2= \frac{25}{16}x^2 [/tex]
Then [tex]E(X^2)=(2x)^2\times \frac{1}{2}+ (\frac{x}{2})^2\times \frac{1}{2}= \frac{17}{8}x^2 [/tex]
Finally the standard deviation:
[tex] \sqrt{\frac{17}{8}x^2- \frac{25}{16}x^2}= \frac{3}{4}x [/tex]
