Answer
$E(X)=0$
Work Step by Step
There are five possible values for the random variable X: a, b, c, d, e. The probability for each one of them to be choosen, if the student just guesses, is equal to $\frac{1}{5}$. That is,
$P(a)=P(b)=P(c)=P(d)=P(e)=\frac{1}{5}$
There is only one correct option, let's say b. So, if the student chooses b he receives 1 point. For any other option, he loses $\frac{1}{4}$ point.
$E(X)=μ_X=Σ[x.P(x)]=(-\frac{1}{4})\times\frac{1}{5}+1\times\frac{1}{5}+(-\frac{1}{4})\times\frac{1}{5}+(-\frac{1}{4})\times\frac{1}{5}+(-\frac{1}{4})\times\frac{1}{5}=1\times\frac{1}{5}-4\times\frac{1}{4}\times\frac{1}{5}=\frac{1}{5}-\frac{1}{5}=0$
