I've got random variable $x$ and probabilities: $P(x=0)=1/4, P(x\in[a,b])=\cfrac{b-a}{2}, P(x=3)=1/4$ where $1\leq a < b \leq 2$.
What I did:
First I tried to calculate the mathematical expectation $M(X)=0\cdot\cfrac{1}{4} + \int\limits_a^b x \cfrac{b-a}{2}dx+3\cdot \cfrac{1}{4}$. But the result seems too complicated for such a simple task. Am I getting anything wrong?
As pointed out in the comment section for expected value we can use $$\mathbb{E}[X] = 0\cdot \frac{1}{4}+ \int_1^2\frac{x}{2}dx+3\cdot\frac{1}{4}$$ Try evaluate it by yourself. For variance I highly recommend you to use $$V(X) = \Bbb E[X^2] - (\Bbb E[X])^2$$ You only need to find $$\mathbb{E}[X^2] = 0^2\cdot \frac{1}{4}+ \int_1^2\frac{x^2}{2}dx+3^2\cdot\frac{1}{4}$$ Hope this helps!