一、(10%) Let $X1, X2$, and $X3$ be independent and have uniform distribution $U(0, 1)$. The joint p.d.f. is $f(x1, x2, x3) = 1$, where 0 x1, x2, x3 1. What is the probability P(X1 + X2 + X3 1)? (Please show the details)

四、(10%) Let $X1, X2, \dots, Xn$ be a random sample from $N(\mu, \sigma^2)$ with known mean $\mu$. Moreover, Let $\chi^2{\mathfrak{p}(n)}$ denote the $100p^n$ percentile of $\chi^2(n)$, the Chi-Square distribution with $n$ degrees of freedom, where $0 \le p \le 1$. Namely, $P(Y \le \chi^2_{\mathfrak{p}(n)}) = p$, where $Y \sim \chi^2(n)$. Describe how you would construct a $100(1-\alpha)\%$ confidence interval for the unknown variance $\sigma^2$, where $0 \le \alpha \le 1$.

一、(10%) Let $X1, X2$, and $X3$ be independent and have uniform distribution $U(0, 1)$. The joint p.d.f. is $f(x1, x2, x3) = 1$, where $0 \le x1, x2, x3 \le 1$. What is the probability $P(X1 + X2 + X3 \le 1)$? (Please show the details)

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