Beta Distribution and Expectation and Variance

Question

Let X ∼ Beta(a, b). Compute E[X] and Var(X).

So confused how to start this question.

Any hint would be helpful.

Answer 1

The support of a Beta distribution is always the interval of $(0;1)$. (Unless shifted or scaled; which you did not asked for.)

The pdf is $X\sim\mathcal{Beta}(\alpha, \beta) \iff f_X(x) = \begin{cases}(x^{\alpha-1} (1-x)^{\beta-1})/(\mathcal B(\alpha, \beta)) & : 0<x<1 \\ 0 & : \text{otherwise}\end{cases}$

Where $\mathcal B(\cdot ,\cdot )$ is the Beta function: $\mathcal B(\alpha, \beta) = \frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)} = \int_0^1 u^{\alpha-1} (1-u)^{\beta-1}\operatorname d u$

Where $\Gamma(\cdot)$ is the Gamma function, the extension of the factorial function to the Reals. ...

Use this to compute $\mathsf E(X) = \int_0^1 x\,f_X(x)\operatorname d x$ and then $\mathsf {Var}(X) = \int_0^1 x^2\, f_X(x)\operatorname d x - E(X)^2$.