Question

Let \(X_1,X_2,X_3,X_4\) be a random sample of size \(4\) from a distribution with the probability density function

Hint:

In the method of moments, equate sample moments with corresponding population moments and solve for the unknown parameter.

Answer 1

Correct Answer: 0.67

Step 1: Sample mean.
The data $0.2,0.1,0.3,0.4$ average to $0.25$.

Step 2: Population mean.
For $f(x)=\alpha(\alpha+1)x^{\alpha-1}(1-x)$ on $(0,1)$, integrating gives $E(X)=\frac\alpha{\alpha+2}$.

Step 3: Match the moments.
Set $\frac\alpha{\alpha+2}=0.25=\frac14$, so $4\alpha=\alpha+2$.

Step 4: Solve.
$3\alpha=2$, so $\alpha=\frac23=0.666\ldots$

Step 5: Round.
\[ \boxed{0.67} \]