{"id":2126,"date":"2023-04-02T09:49:46","date_gmt":"2023-04-02T01:49:46","guid":{"rendered":"https:\/\/www.appblog.cn\/?p=2126"},"modified":"2023-04-05T20:41:58","modified_gmt":"2023-04-05T12:41:58","slug":"fundamentals-of-high-school-mathematics-random-variables-and-their-distributions","status":"publish","type":"post","link":"https:\/\/www.appblog.cn\/index.php\/2023\/04\/02\/fundamentals-of-high-school-mathematics-random-variables-and-their-distributions\/","title":{"rendered":"\u9ad8\u4e2d\u6570\u5b66\u57fa\u7840\uff1a\u968f\u673a\u53d8\u91cf\u53ca\u5176\u5206\u5e03"},"content":{"rendered":"

\u57fa\u672c\u6982\u5ff5<\/h2>\n

\u82e5\u968f\u673a\u53d8\u91cf$X$\u670d\u4ece\u4e00\u4e2a\u6570\u5b66\u671f\u671b\u4e3a$\u03bc$\u3001\u65b9\u5dee\u4e3a$\u03c3^2$\u7684\u6b63\u6001\u5206\u5e03\uff0c\u8bb0\u4e3a$N(\u03bc\uff0c\u03c3^2)$\u3002\u5176\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u4e3a\u6b63\u6001\u5206\u5e03\u7684\u671f\u671b\u503c$\u03bc$\u51b3\u5b9a\u4e86\u5176\u4f4d\u7f6e\uff0c\u5176\u6807\u51c6\u5dee$\u03c3$\u51b3\u5b9a\u4e86\u5206\u5e03\u7684\u5e45\u5ea6\u3002\u5f53$\u03bc = 0, \u03c3 = 1$\u65f6\u7684\u6b63\u6001\u5206\u5e03\u662f\u6807\u51c6\u6b63\u6001\u5206\u5e03\u3002<\/p>\n


\n\"\u9ad8\u4e2d\u6570\u5b66\u57fa\u7840_\u968f\u673a\u53d8\u91cf\"<\/p>\n

\u79bb\u6563\u578b\u968f\u673a\u53d8\u91cf<\/h2>\n

\uff081\uff09\u968f\u673a\u53d8\u91cf<\/strong>\uff1a\u5982\u679c\u968f\u673a\u8bd5\u9a8c\u7684\u7ed3\u679c\u53ef\u4ee5\u7528\u4e00\u4e2a\u53d8\u91cf\u6765\u8868\u793a\uff0c\u90a3\u4e48\u8fd9\u6837\u7684\u53d8\u91cf\u53eb\u505a\u968f\u673a\u53d8\u91cf\u3002\u968f\u673a\u53d8\u91cf\u5e38\u7528\u5b57\u6bcd$X,Y,\\xi,\\eta$ \u7b49\u8868\u793a\u3002<\/p>\n

\uff082\uff09\u79bb\u6563\u578b\u968f\u673a\u53d8\u91cf<\/strong>\uff1a\u5bf9\u4e8e\u968f\u673a\u53d8\u91cf\u53ef\u80fd\u53d6\u7684\u503c\uff0c\u53ef\u4ee5\u6309\u4e00\u5b9a\u6b21\u5e8f\u4e00\u4e00\u5217\u51fa\uff0c\u8fd9\u6837\u7684\u968f\u673a\u53d8\u91cf\u53eb\u505a\u79bb\u6563\u578b\u968f\u673a\u53d8\u91cf\u3002<\/p>\n

\uff083\uff09\u8fde\u7eed\u578b\u968f\u673a\u53d8\u91cf<\/strong>\uff1a\u5bf9\u4e8e\u968f\u673a\u53d8\u91cf\u53ef\u80fd\u53d6\u7684\u503c\uff0c\u53ef\u4ee5\u53d6\u67d0\u4e00\u533a\u95f4\u5185\u7684\u4e00\u5207\u503c\uff0c\u8fd9\u6837\u7684\u53d8\u91cf\u5c31\u53eb\u505a\u8fde\u7eed\u578b\u968f\u673a\u53d8\u91cf\u3002<\/p>\n

\uff084\uff09\u79bb\u6563\u578b\u968f\u673a\u53d8\u91cf\u4e0e\u8fde\u7eed\u578b\u968f\u673a\u53d8\u91cf\u7684\u533a\u522b\u4e0e\u8054\u7cfb\uff1a\u79bb\u6563\u578b\u968f\u673a\u53d8\u91cf\u4e0e\u8fde\u7eed\u578b\u968f\u673a\u53d8\u91cf\u90fd\u662f\u7528\u53d8\u91cf\u8868\u793a\u968f\u673a\u8bd5\u9a8c\u7684\u7ed3\u679c\uff1b\u4f46\u662f\u79bb\u6563\u578b\u968f\u673a\u53d8\u91cf\u7684\u7ed3\u679c\u53ef\u4ee5\u6309\u4e00\u5b9a\u6b21\u5e8f\u4e00\u4e00\u5217\u51fa\uff0c\u800c\u8fde\u7eed\u578b\u968f\u673a\u53d8\u91cf\u7684\u7ed3\u679c\u4e0d\u53ef\u4ee5\u4e00\u4e00\u5217\u51fa\u3002<\/p>\n

\uff085\uff09\u82e5$X$\u662f\u968f\u673a\u53d8\u91cf\uff0c$Y=aX+b$\uff08$a,b$\u662f\u5e38\u6570\uff09\uff0c\u5219$Y$\u4e5f\u662f\u968f\u673a\u53d8\u91cf\uff0c\u5e76\u4e14\u4e0d\u6539\u53d8\u5176\u5c5e\u6027\uff08\u79bb\u6563\u578b\u3001\u8fde\u7eed\u578b\uff09\u3002<\/p>\n

\u79bb\u6563\u578b\u968f\u673a\u53d8\u91cf\u7684\u5206\u5e03\u5217<\/h2>\n

\u6982\u7387\u5206\u5e03\uff08\u5206\u5e03\u5217\uff09<\/h3>\n

\u8bbe\u79bb\u6563\u578b\u968f\u673a\u53d8\u91cf$X$\u53ef\u80fd\u53d6\u7684\u4e0d\u540c\u503c\u4e3a$x_1,x_2,\u00b7\u00b7\u00b7,x_i,\u00b7\u00b7\u00b7,x_n$\uff0c$X$\u7684\u6bcf\u4e00\u4e2a\u503c$x_i \\, (i=1,2,\u00b7\u00b7\u00b7,n)$\u7684\u6982\u7387$P(X=x_i)=p_i$\uff0c\u5219\u79f0\u8868<\/p>\n\n\n\n\n\n
$X$<\/th>\n$x_1$<\/th>\n$x_2$<\/th>\n$\u00b7\u00b7\u00b7$<\/th>\n$x_i$<\/th>\n$\u00b7\u00b7\u00b7$<\/th>\n$x_n$<\/th>\n<\/tr>\n<\/thead>\n
$P$<\/td>\n$p_1$<\/td>\n$p_2$<\/td>\n$\u00b7\u00b7\u00b7$<\/td>\n$p_i$<\/td>\n$\u00b7\u00b7\u00b7$<\/td>\n$p_n$<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

\u4e3a\u968f\u673a\u53d8\u91cf$X$\u7684\u6982\u7387\u5206\u5e03\uff0c\u7b80\u79f0$X$\u7684\u5206\u5e03\u5217\u3002<\/p>\n

\u6027\u8d28\uff1a<\/p>\n

\u2460 $pi\u22650,i=1,2,…n$
\n\u2461 $\\sum<\/em>{i=1}^np_i=1$<\/p>\n

\u4e24\u70b9\u5206\u5e03<\/h3>\n

\u5982\u679c\u968f\u673a\u53d8\u91cf$X$\u7684\u5206\u5e03\u5217\u4e3a<\/p>\n\n\n\n\n\n
$X$<\/th>\n0<\/th>\n1<\/th>\n<\/tr>\n<\/thead>\n
$P$<\/td>\n$1-p$<\/td>\n$p$<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

\u5219\u79f0$X$\u670d\u4ece\u4e24\u70b9\u5206\u5e03<\/strong>\uff0c\u5e76\u79f0$p=P(X=1)$\u4e3a\u6210\u529f\u5206\u5e03\u3002<\/p>\n

\u4e8c\u9879\u5206\u5e03<\/h3>\n

\u5982\u679c\u5728\u4e00\u6b21\u8bd5\u9a8c\u4e2d\u67d0\u4e8b\u4ef6\u53d1\u751f\u7684\u6982\u7387\u662f$p$\uff0c\u90a3\u4e48\u5728$n$\u6b21\u72ec\u7acb\u91cd\u590d\u8bd5\u9a8c\u4e2d\u8fd9\u4e2a\u4e8b\u4ef6\u6070\u597d\u53d1\u751f$k$\u6b21\u7684\u6982\u7387\u662f<\/p>\n

$P(X=k) = C_n^kp^k(1-p)^{n-k}$\uff0c\u5176\u4e2d$k=0,1,2,…,n, \\, q=1-p$\uff0c\u4e8e\u662f\u5f97\u5230\u968f\u673a\u53d8\u91cf$X$\u7684\u6982\u7387\u5206\u5e03\u5982\u4e0b\uff1a<\/p>\n\n\n\n\n\n
$X$<\/th>\n0<\/th>\n1<\/th>\n$\u00b7\u00b7\u00b7$<\/th>\n$k$<\/th>\n$\u00b7\u00b7\u00b7$<\/th>\n$n$<\/th>\n<\/tr>\n<\/thead>\n
$P$<\/td>\n$C_n^0p^0q^n$<\/td>\n$C_n^1p^1q^{n-1}$<\/td>\n$\u00b7\u00b7\u00b7$<\/td>\n$C_n^kp^kq^{n-k}$<\/td>\n$\u00b7\u00b7\u00b7$<\/td>\n$C_n^np^nq^0$<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

\u6211\u4eec\u79f0\u8fd9\u6837\u7684\u968f\u673a\u53d8\u91cf$X$\u670d\u4ece\u4e8c\u9879\u5206\u5e03<\/strong>\uff0c\u8bb0\u4f5c$X~B(n,p)$\uff0c\u5e76\u79f0$p$\u4e3a\u6210\u529f\u6982\u7387\u3002<\/p>\n

\u5224\u65ad\u4e00\u4e2a\u968f\u673a\u53d8\u91cf\u662f\u5426\u670d\u4ece\u4e8c\u9879\u5206\u5e03\uff0c\u5173\u952e\u6709\u4e09\u70b9\uff1a
\n\u2460 \u72ec\u7acb\u6027\uff1a\u5373\u4e00\u6b21\u8bd5\u9a8c\u4e2d\u4e8b\u4ef6\u53d1\u751f\u4e0e\u5426\u4e8c\u8005\u5fc5\u5c45\u5176\u4e00\uff1b
\n\u2461 \u91cd\u590d\u6027\uff1a\u5373\u8bd5\u9a8c\u662f\u72ec\u7acb\u91cd\u590d\u5730\u8fdb\u884c\u4e86$n$\u6b21
\n\u2462 \u7b49\u6982\u7387\u6027\uff1a\u5728\u6bcf\u6b21\u8bd5\u9a8c\u4e2d\u4e8b\u4ef6\u53d1\u751f\u7684\u6982\u7387\u5747\u76f8\u7b49\u3002<\/p>\n

\u6ce8\uff1a\u4e8c\u9879\u5206\u5e03\u7684\u6a21\u578b\u662f\u6709\u653e\u56de\u62bd\u6837<\/p>\n

\u79bb\u6563\u578b\u968f\u673a\u53d8\u91cf\u7684\u5747\u503c\u4e0e\u65b9\u5dee<\/h3>\n

\uff081\uff09\u79bb\u6563\u578b\u968f\u673a\u53d8\u91cf\u7684\u5747\u503c<\/p>\n

\u4e00\u822c\u5730\uff0c\u82e5\u79bb\u6563\u578b\u968f\u673a\u53d8\u91cf$X$\u7684\u5206\u5e03\u5217\u4e3a<\/p>\n\n\n\n\n\n
$X$<\/th>\n$x_1$<\/th>\n$x_2$<\/th>\n$\u00b7\u00b7\u00b7$<\/th>\n$x_i$<\/th>\n$\u00b7\u00b7\u00b7$<\/th>\n$x_n$<\/th>\n<\/tr>\n<\/thead>\n
$P$<\/td>\n$p_1$<\/td>\n$p_2$<\/td>\n$\u00b7\u00b7\u00b7$<\/td>\n$p_i$<\/td>\n$\u00b7\u00b7\u00b7$<\/td>\n$p_n$<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

\u5219\u79f0\uff1a<\/p>\n

$E(X)=x_1p_1+x_2p_2+\u00b7\u00b7\u00b7+x_ip_i+\u00b7\u00b7\u00b7+x_np_n$<\/p>\n

\u4e3a\u79bb\u6563\u578b\u968f\u673a\u53d8\u91cf$X$\u7684\u5747\u503c<\/strong>\u6216\u671f\u671b<\/strong>\uff0c\u53cd\u6620\u4e86\u79bb\u6563\u578b\u968f\u673a\u53d8\u91cf\u53d6\u503c\u7684\u5e73\u5747\u6c34\u5e73\u3002<\/p>\n

\u6027\u8d28\uff1a<\/p>\n

\u2460 $E(aX+b)=aE(X)+b$
\n\u2461 \u82e5$X$\u670d\u4ece\u4e24\u70b9\u5206\u5e03\uff0c\u5219$E(X)=p$
\n\u2462 \u82e5$X~B(n,p)$\uff0c\u5219$E(X)=np$<\/p>\n

\uff082\uff09\u79bb\u6563\u578b\u968f\u673a\u53d8\u91cf\u7684\u65b9\u5dee<\/p>\n

\u4e00\u822c\u5730\uff0c\u82e5\u79bb\u6563\u578b\u968f\u673a\u53d8\u91cf$X$\u7684\u5206\u5e03\u5217\u4e3a<\/p>\n\n\n\n\n\n
$X$<\/th>\n$x_1$<\/th>\n$x_2$<\/th>\n$\u00b7\u00b7\u00b7$<\/th>\n$x_i$<\/th>\n$\u00b7\u00b7\u00b7$<\/th>\n$x_n$<\/th>\n<\/tr>\n<\/thead>\n
$P$<\/td>\n$p_1$<\/td>\n$p_2$<\/td>\n$\u00b7\u00b7\u00b7$<\/td>\n$p_i$<\/td>\n$\u00b7\u00b7\u00b7$<\/td>\n$p_n$<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

\u5219\u79f0\uff1a<\/p>\n

$D(X)=\\sum_{i=1}^n(x_i-E(X))^2p_i$<\/p>\n

\u4e3a\u79bb\u6563\u578b\u968f\u673a\u53d8\u91cf$X$\u7684\u65b9\u5dee<\/strong>\uff0c\u5e76\u79f0\u5176\u7b97\u672f\u5e73\u65b9\u6839$\\sqrt{D(X)}$\u4e3a\u968f\u673a\u53d8\u91cf$X$\u7684\u6807\u51c6\u5dee\u3002\u53cd\u6620\u4e86\u79bb\u6563\u578b\u968f\u673a\u53d8\u91cf\u53d6\u503c\u7684\u7a33\u5b9a\u4e0e\u6ce2\u52a8\uff0c\u96c6\u4e2d\u4e0e\u79bb\u6563\u7684\u7a0b\u5ea6\u3002$D(X)$\u8d8a\u5c0f\uff0c$X$\u7684\u7a33\u5b9a\u6027\u8d8a\u9ad8\uff0c\u6ce2\u52a8\u8d8a\u5c0f\uff0c\u53d6\u503c\u8d8a\u96c6\u4e2d\uff1b$D(X)$\u8d8a\u5927\uff0c$X$\u7684\u7a33\u5b9a\u6027\u8d8a\u5dee\uff0c\u6ce2\u52a8\u8d8a\u5927\uff0c\u53d6\u503c\u8d8a\u5206\u6563\u3002<\/p>\n

\u2460 $D(aX+b)=a^2D(X)$
\n\u2461 \u82e5$X$\u670d\u4ece\u4e24\u70b9\u5206\u5e03\uff0c\u5219$D(X)=p(1-p)$
\n\u2462 \u82e5$X~B(n,p)$\uff0c\u5219$D(X)=np(1-p)$<\/p>\n

\u6b63\u6001\u5206\u5e03<\/h2>\n

\u6b63\u6001\u5206\u5e03\u6982\u7387\u5bc6\u5ea6\u66f2\u7ebf\u51fd\u6570\u8868\u8fbe\u5f0f\u4e3a\uff1a<\/p>\n

$$
\nf(x)=\\frac{1}{\\sqrt{2\\pi}\u00b7\\sigma}e^{-\\frac{(x-\\mu)^2}{2\\sigma^2}}, \\, x \\in R
\n$$<\/p>\n

\u5176\u4e2d$\\mu,\\sigma$\u662f\u53c2\u6570\uff0c\u4e14$\\sigma>0,-\u221e<\\mu<+\u221e$\uff0c\u8bb0\u4f5c$N(\\mu,\\sigma^2)$\uff0c\u66f2\u7ebf\u5982\u4e0b\u56fe\uff1a<\/p>\n

\"\u9ad8\u4e2d\u6570\u5b66\u57fa\u7840_\u6b63\u6001\u5206\u5e03\"<\/p>\n","protected":false},"excerpt":{"rendered":"

\u57fa\u672c\u6982\u5ff5 \u82e5\u968f\u673a\u53d8\u91cf$X$\u670d\u4ece\u4e00\u4e2a\u6570\u5b66\u671f\u671b\u4e3a$\u03bc$\u3001\u65b9\u5dee\u4e3a$\u03c3^2$\u7684\u6b63\u6001\u5206\u5e03\uff0c\u8bb0\u4e3a$N(\u03bc\uff0c\u03c3^2)$\u3002\u5176\u6982 […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[509],"tags":[],"class_list":["post-2126","post","type-post","status-publish","format-standard","hentry","category-mathematics-fundamentals"],"_links":{"self":[{"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/posts\/2126","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/comments?post=2126"}],"version-history":[{"count":0,"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/posts\/2126\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/media?parent=2126"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/categories?post=2126"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/tags?post=2126"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}