Beta Distribution: Derivation, Moments, and Connections

Consider the Beta distribution with PDF $f(x) = \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)} x^{\alpha-1}(1-x)^{\beta-1}$ for $x \in (0,1)$.

(a) Verify that $f$ integrates to 1 by showing $\int_0^1 x^{\alpha-1}(1-x)^{\beta-1}\,dx = \frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)}$ using the integral representation of the Gamma function.

(b) Derive $E[X] = \frac{\alpha}{\alpha+\beta}$ and $\text{Var}(X) = \frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}$.

(c) Show that $\text{Beta}(1,1)$ reduces to $\text{Uniform}(0,1)$.

(d) If $Y_1 \sim \text{Gamma}(\alpha, 1)$ and $Y_2 \sim \text{Gamma}(\beta, 1)$ are independent, explain (without full proof) why $\frac{Y_1}{Y_1+Y_2} \sim \text{Beta}(\alpha, \beta)$.