1. (10%) Suppose X_1, X_2 are independent and both follow continuous uniform distribution U(0,1). Please find the probability Pr(X_1^2 + X_2^2 &lt; 1).

3. (10%) Let \( X_1, X_2, \dots, X_n \) be a random sample of size n from the distribution with p.d.f. \( f(x, \theta) = \theta x^{\theta-1} \), \( 0 < x < 1 \), \( 0 < \theta < \infty \). Assume that \( \overline{X} = \sum_{i=1}^{n} X_i / n = 0.2 \) be the sample mean. Please use \( \overline{X} \) to find the moments estimator of \( \theta \) and calculate its estimate.

1. (10%) Suppose \( X_1 \), \( X_2 \) are independent and both follow continuous uniform distribution U(0,1). Please find the probability \( Pr(X_1^2 + X_2^2 < 1) \).

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