![\"\"](\"https:\/\/egonlin.com\/wp-content\/uploads\/2022\/02\/\u5e38\u89c1\u7684\u6982\u7387\u5206\u5e03\u6a21\u578b-\u8054\u5408\u5206\u5e03\u51fd\u6570.png\")
<\/div><\/p>\n\u591a\u9879\u5206\u5e03\uff08Multinomial Distribution\uff09<\/h1>\n\u591a\u9879\u5206\u5e03\u7b80\u4ecb<\/h2>\n
\u591a\u9879\u5206\u5e03\u662f\u4e8c\u9879\u5206\u5e03\u7684\u63a8\u5e7f\uff0c\u4ed6\u4eec\u7684\u533a\u522b\u662f\u4e8c\u9879\u5206\u5e03\u7684\u7ed3\u679c\u53ea\u6709$0$\u548c$1$\u4e24\u79cd\uff0c\u591a\u9879\u5f0f\u7684\u7ed3\u679c\u53ef\u4ee5\u6709\u591a\u4e2a\u503c\u3002<\/p>\n
\u591a\u9879\u5206\u5e03\u7684\u5178\u578b\u4f8b\u5b50\u662f\u63b7\u9ab0\u5b50\uff0c6\u4e2a\u70b9\u5bf9\u5e946\u4e2a\u4e0d\u540c\u7684\u6570\uff0c\u6bcf\u4e2a\u70b9\u7684\u6982\u7387\u90fd\u4e3a${\\frac{1}{6}}$<\/p>\n
\u4e0e\u4e8c\u9879\u5206\u5e03\u7c7b\u4f3c\uff0c\u591a\u9879\u5206\u5e03\u6765\u81ea\u4e8e$(p_1+p_2+\\cdots+p_k)^n\u591a\u9879\u5f0f\u7684\u5c55\u5f00$<\/p>\n
\u591a\u9879\u5206\u5e03\u516c\u5f0f\u89e3\u6790<\/h2>\n
\u4ee5\u63b7\u9ab0\u5b50\u4e3a\u4f8b\uff0c\u63b7\u9ab0\u5b50\u7684\u65f6\u5019\u63b7$1-6$\u7684\u6982\u7387\u90fd\u4e3a${\\frac{1}{6}}$\uff0c\u8bb0\u4f5c$p_1-p_6$\uff0c\u53ef\u4ee5\u53d1\u73b0$p_1+p_2+p_3+p_4+p_5+p_6=1$\uff0c\u73b0\u5728\u628a$p_1+p_2+p_3+p_4+p_5+p_6$\u8bb0\u4f5c\u505a\u4e00\u6b21\u62bd\u6837\u5404\u79cd\u4e8b\u4ef6\u53d1\u751f\u7684\u6982\u7387\u548c\uff0c\u5373\u53ef\u5f97$(p_1+p_2+p_3+p_4+p_5+p_6)^n=1^n$\u4e3a$n$\u6b21\u62bd\u6837\u6240\u6709\u4e8b\u4ef6\u76f8\u4e92\u7ec4\u5408\u5bf9\u5e94\u7684\u6982\u7387\u548c\uff0c\u4e4b\u540e\u4f7f\u7528\u591a\u9879\u5f0f\u5c55\u5f00(\u6ce8\uff1a\u4f7f\u7528\u591a\u9879\u5f0f\u5b9a\u7406\u5c55\u5f00\uff0c\u7531\u4e8e\u591a\u9879\u5f0f\u5b9a\u7406\u4e0d\u5728\u672c\u8282\u63d0\u53ca\u8303\u56f4\u5185\uff0c\u4e0d\u591a\u8d58\u8ff0)\uff0c\u5982\u679c\u5b83\u4e0d\u662f\u63b7\u9ab0\u5b50\uff0c\u800c\u662f\u4e00\u4e2a\u6709$n$\u79cd\u53ef\u80fd\u7684\u95ee\u9898\uff0c\u4f1a\u5f97\u5230\u4e00\u4e2a\u591a\u9879\u5f0f\u5c55\u5f00\u7684\u516c\u5f0f \u4f2f\u52aa\u5229\u5206\u5e03\u662f\u4e00\u4e2a\u4e8c\u503c\u79bb\u6563\u5206\u5e03\uff0c\u7ed3\u679c\u53ea\u6709$0$\u548c$1$\u4e24\u79cd\u3002<\/p>\n \u968f\u5373\u53d8\u91cf$X$\u4e3a$1$\u7684\u6982\u7387\u4e3a$p$\uff0c\u5219\u4e3a$0$\u7684\u6982\u7387\u4e3a$q=1-p$\uff0c\u53ef\u4ee5\u7528\u516c\u5f0f\u8868\u793a\u4e3a \u4f2f\u52aa\u5229\u5206\u5e03\u7684\u671f\u671b\u503c\u4e3a <\/p> \u8054\u7cfb\u7ba1\u7406\u5458\u5fae\u4fe1tutu19192010\uff0c\u6ce8\u518c\u8d26\u53f7<\/p>\n<\/div>
\n$$
\nP(X_1 = x_1,\\ldots,X_k = x_k) = \\begin{cases}
\n{\\frac{n!}{x_1!\\cdots{x_k!}}}(p^{x_1}\\cdots{p^{xk})} \\quad when\\sum<\/em>{i=1}^kx_i=n\\
\n0 \\quad otherwise \\
\n\\end{cases}
\n$$
\n\u8fd9\u4e2a\u591a\u9879\u5f0f\u8868\u793a$X_1$\u51fa\u73b0$x_1$\u6b21\uff0c$X_2$\u51fa\u73b0$x_2$\u6b21\uff0c$\\ldots$\uff0c$X_k$\u51fa\u73b0$x_k$\u6b21\u7684\u51fa\u73b0\u6982\u7387\uff0c\u8fd9\u6837\u5c31\u5f97\u5230\u4e86\u4e0a\u8ff0\u6240\u793a\u7684\u591a\u9879\u5206\u5e03\u7684\u591a\u9879\u5c55\u5f00\u5f0f\u516c\u5f0f\u3002<\/p>\n\u4f2f\u52aa\u5229\u5206\u5e03\uff08Bernoulli Distribution\uff09<\/h1>\n
\u4f2f\u52aa\u5229\u5206\u5e03\u7b80\u4ecb<\/h2>\n
\n$$
\nf(x) = p^x(1-p)^{1-x} = \\begin{cases}
\np, \\quad\\quad x=1 \\
\n1-p, \\quad x=0 \\
\n\\end{cases}
\n$$<\/p>\n\u4f2f\u52aa\u5229\u5206\u5e03\u7684\u671f\u671b\u503c\u548c\u65b9\u5dee<\/h2>\n
\n$$
\n\\begin{align}
\nE(X) & = \\sum_{i=0}^1xif(x) \\
\n& = 1p+0<\/em>(1-p) \\
\n& = p+0 \\
\n& = p \\
\n\\end{align}
\n$$
\n \u4f2f\u52aa\u5229\u5206\u5e03\u7684\u65b9\u5dee\u4e3a
\n$$
\n\\begin{align}
\nD(x) & = \\sum<\/em>{i=0}^1(x_i – E(x))^2f(x) \\
\n& = (1-E(x))^2p + (0-E(x)^2<\/em>(1-p) \\
\n& = (1-p)^2p + (0-p)^2<\/em>(1-p) \\
\n& = p – p^2 \\
\n& = p(1-p) \\
\n& = pq
\n\\end{align}
\n$$<\/p>\n\u6b63\u6001(\u9ad8\u65af)\u5206\u5e03\uff08Normal(Gaussian) Distribution\uff09<\/h1>\n
\u6b63\u6001\u5206\u5e03\u7684\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u56fe\u50cf<\/h2>\n
\u67e5\u770b\u66f4\u591a<\/h2>