Independent solution

How to solve this Beta Distribution question

Setup

Setup

Match the density kernel to a beta distribution.

f(x)=20x4−1(1−x)2−1f(x)=20x^{4-1}(1-x)^{2-1}
X∼Beta⁡(4,2)X\sim\operatorname{Beta}(4,2)

Model

Model

Use the beta variance formula before taking a square root.

Var⁡(X)=αβ(α+β)2(α+β+1)\operatorname{Var}(X)=\frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}

Compute

Compute

Substitute alpha=4 and beta=2.

Var⁡(X)=4⋅262⋅7=263\operatorname{Var}(X)=\frac{4\cdot2}{6^2\cdot7}=\frac{2}{63}
SD⁡(X)=263=0.1781741613\operatorname{SD}(X)=\sqrt{\frac{2}{63}}=0.1781741613

Answer

Answer

The standard deviation rounds to 0.178.

0.178(B)\boxed{0.178\quad\text{(B)}}