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zV3PjJ@(FmT{mL?aFbQ#hL0H=XuV%ov-T_JRmz{S_3z!?s5#ZrwFknv}zLox$^d({8 z2#9s?3CJc7hA&S5M*%YUi{Ze%;(jh$e=%0LYu!(fJkXmI3@}^9UsZ6XMcl%*m)uUpcw+<{s+wvz$fd^v>-6xllVt|AWry+%x`6Ia6;e$ z!EZ1S2m**Wf55>1Yy%E15a5~lJ01rYnCH(jI61g@{;V4Z7cX2X_^n(HK(zX^zMMQ@ zxS;0fTEO@SnfO9}gGc1D2{iLR>j42|xj*ycg#ezm|0%=S5o%!#b42| [ + #set align(center) + + #if lev == 1 { + text(size: 22pt, font: "Source Han Serif SC")[#it.body] + } else if lev == 2 { + text(size: 18pt, font: "FZXiaoBiaoSong-B05S")[#it.body] + } + ] + heading(body, level: lev) +} + +#title(lev: 1)[概率论(H)2023-2024 秋冬小测] + +#align(center)[#text(size: 15pt,font: "Ink Free",[shrike505])] + +#title(lev: 2)[第一次小测:2024年10月11日] + ++ + + 假设事件 $A, B, C$ 互不相容,且 $P(A) = P(B) = 0.2, P(C) = 0.4$. 求 $P(A dash.en(C) union B)$. + + 假设事件 $A, B$ 相互独立,且 $P(A) = 0.6, P(B) = 0.3$. 求 $A$ 与 $B$ 至少有一个发生的条件下 $A$ 发生的概率. + + 对随机事件 $A, B$,若 $P(A bar B) = 0.2, P(dash.en(A) bar dash.en(B)) = 0.3, P(B) = 0.4$,求 $P(B bar A)$. ++ 在线段 $A B$ 中随机取三点 $C, D, E$, 求三线段 $A C, A D, A E$ 可构成三角形的概率. ++ 投票选举甲乙两人,已知甲共得 $m$ 张,乙共得 $n$ 张,$m gt.eq n + 2$. 求在第一张票后的计票过程中,甲得票数始终至少领先乙得票数 $2$ 张的概率. + +#title(lev: 2)[第二次小测:2024年11月22日] + ++ 设随机变量 $xi_1, xi_2, dots, xi_n$ 相互独立,且同分布. $P(xi_1 = i) = frac(2, 3^i), i = 1, 2, dots, n$. 令 $eta = min_(1 lt.eq k lt.eq n) xi_k$ + + 求 $P(eta gt.eq i)$. + + 求 $eta$ 的分布列. ++ 设 $X, Y$ 为非负整数值随机变量,$X + Y tilde P(lambda)$(泊松分布),并在给定 $X + Y = n$ 的条件下,$X tilde B(n, p)$. 求 $X, Y$ 的分布. ++ 设随机变量 $X_1, X_2, X_3$ 相互独立,都服从参数为 1 的指数分布, + + 记 $Y_1 = frac(X_1, X_1 + X_2), Y_2 = frac(X_1 + X_2, X_1 + X_2 + X_3), Y_3 = X_1 + X_2 + X_3$. + + + 求 $(Y_1, Y_2, Y_3)$ 的联合密度函数. + + 证明 $Y_1, Y_2, Y_3$ 相互独立. \ No newline at end of file