數(shù)學(xué)期望公開課

猜猜我是誰姚明一個(gè)人的【特征】可以幫助我們記住一個(gè)人幫助我們判斷一個(gè)人幫助我們?cè)u(píng)價(jià)一個(gè)人引例1某人射擊10次,命中的環(huán)數(shù)環(huán)數(shù)分別是:1，1，1，1，2，2，2，3，3，4；則所得的平均環(huán)數(shù)是多少？把環(huán)數(shù)看成隨機(jī)變量，則其分布律為：引例2某商場要將單價(jià)分別為18元/kg，24元/kg，36元/kg的3種糖果按3：2：1的比例混合銷售，如何對(duì)混合糖果定價(jià)才合理？182436把三種糖果的單價(jià)看成隨機(jī)變量，則其分布律為：離散型隨機(jī)變量的數(shù)學(xué)期望mathematicalexpectation

隨機(jī)變量及數(shù)字特征

注意點(diǎn)數(shù)學(xué)期望簡稱為期望數(shù)學(xué)期望又稱為均值數(shù)學(xué)期望是一種加權(quán)平均值數(shù)學(xué)期望是常數(shù)，不是變量能夠刻畫隨機(jī)變量某些性質(zhì)特征的量稱為隨機(jī)變量的數(shù)字特征，數(shù)學(xué)期望是隨機(jī)變量的重要數(shù)字特征數(shù)學(xué)期望是隨機(jī)變量的基本特征之一例1某人在商場購物滿300元，獲得抽獎(jiǎng)資格。具體得獎(jiǎng)多少需要自己由“大轉(zhuǎn)盤轉(zhuǎn)出。大轉(zhuǎn)盤共設(shè)100格，期中8格為每格標(biāo)500元；另10格每格80元；又20格為每格10元；又30格為每格5元；其余另32格無獎(jiǎng)。問：他能得到獎(jiǎng)金的期望是多少？例題講解用定義求隨機(jī)變量均值的一般步驟：1、求出隨機(jī)變量所有可能取值2、求出分布列3、利用數(shù)學(xué)期望的定義求均值例2、隨機(jī)變量X的分布列是(1)a=?(2)E(2X+1)=？(3)

？

例3

選拔運(yùn)動(dòng)員

設(shè)某教練員有甲、乙兩名射擊運(yùn)動(dòng)員,現(xiàn)需要選拔其中的一名參加運(yùn)動(dòng)會(huì),根據(jù)過去的記錄顯示,二人的技術(shù)水平如下:甲射手乙射手試問哪個(gè)射手技術(shù)較好?解運(yùn)動(dòng)員的水平是通過其平均水平來衡量的,故，甲射手的技術(shù)比較好.因而甲、乙兩射手的平均水平分別為甲：8x0.3+9x0.1+10X0.6=9.3(環(huán)）乙：8x0.2+9x0.5+10x0.3=9.1（環(huán)）

數(shù)學(xué)期望的作用幫助決策數(shù)學(xué)期望是反映隨機(jī)變量總體取值平均水平的一個(gè)重要的數(shù)字特征,而在現(xiàn)實(shí)社會(huì)中由于不確定因素太多，因此人們?cè)谧鰶Q策時(shí)就會(huì)相當(dāng)謹(jǐn)慎，常常會(huì)在多個(gè)決策中找出最好的一個(gè)方案。數(shù)學(xué)期望則成為了決策者們首選的一個(gè)幫助決策的科學(xué)方法。方案決策的步驟1、確定決策目標(biāo)，設(shè)計(jì)備選方案2、分析各可行方案出現(xiàn)的狀態(tài)及概率3、根據(jù)以上數(shù)據(jù)，列出各方案的統(tǒng)計(jì)分布4、數(shù)學(xué)期望幫助決策求各方案隨機(jī)變量的分布列比較各方案的數(shù)學(xué)期望，根據(jù)問題實(shí)際意義，擇優(yōu)選擇求各方案隨機(jī)變量的所有可能取值及對(duì)應(yīng)概率難點(diǎn)例5某投資者有10萬元,現(xiàn)有兩種投資方案:一是投資理財(cái)寶,二是存入銀行獲取利息。理財(cái)寶的收益主要取決于經(jīng)濟(jì)形勢(shì),假設(shè)可分三種狀態(tài):形勢(shì)好(獲利30000元)形勢(shì)中等(獲利10000元)、形勢(shì)不好(損失20000元)。如果存入銀行,假設(shè)年利率6%,即可得利息6000元。解設(shè)X為投資理財(cái)寶收益，Y為存入銀行收益

方案1投資理財(cái)寶：方案2存入銀行：E(X)=30000X0.3+10000X0.5+(-20000)X0.2=10000E(Y)=6000X0.3+6000X0.5+6000X0.2=6000E（X)<E(Y),故應(yīng)選擇投資理財(cái)寶

投資理財(cái)問題乒乓球是我們的國球，中國隊(duì)在這項(xiàng)運(yùn)動(dòng)中具有絕對(duì)的優(yōu)勢(shì)。現(xiàn)就乒乓球比賽的安排提出一個(gè)問題參賽隊(duì)伍：中國隊(duì)—德國隊(duì)賽制一：雙方各出3人，三場兩勝制賽制二：雙方各出5人，五場三勝制哪一種賽制對(duì)中國更有利？

體育比賽問題分析我們利用數(shù)學(xué)期望解答這種決策性的問題由于中國隊(duì)在這項(xiàng)比賽中的優(yōu)勢(shì)，我們不妨設(shè)中國隊(duì)中每一位隊(duì)員對(duì)德國隊(duì)員的勝率都為60%。根據(jù)前面的分析，我們只需要比較中國隊(duì)兩種賽制對(duì)應(yīng)的數(shù)學(xué)期望即可。設(shè)隨機(jī)變量X為該賽制下中國隊(duì)員在比賽中獲勝的場數(shù)，則可建立X的分布律：方案1五場三勝制方案2三場兩勝制設(shè)隨機(jī)變量Y為該賽制下中國隊(duì)員在比賽中獲勝的場數(shù)，則可建立Y的分布律：應(yīng)用離散型隨機(jī)變量X的數(shù)學(xué)期望E(X)2個(gè)概念2個(gè)步驟期望為我們提供了實(shí)際問題決策的理論依據(jù)。用定義數(shù)學(xué)求期望的步驟方法求期望的兩種方法離散型隨機(jī)變量X的函數(shù)Y=f(X)的數(shù)學(xué)期望E[f(X)]決策方案的步驟課堂小結(jié)作業(yè)：P150習(xí)題1719練習(xí)

某工程隊(duì)完成某項(xiàng)工程的時(shí)間為X（單位：月）是一個(gè)隨機(jī)變量，它的分布列為（1）試求該工程隊(duì)完成此項(xiàng)任務(wù)的平均月數(shù)；（2）社該工程隊(duì)所獲利潤為為Y=50(13-X)，單位為萬元，試求該施工隊(duì)的平均利潤練習(xí)

設(shè)袋中有6個(gè)球，其中四個(gè)白球，2個(gè)黑球，從中任取3個(gè)，求抽到的黑球數(shù)X的數(shù)學(xué)期望ClickToTypeSlideTitleHowDoIEditDiagrams?Toeditdiagrams,youshouldfirstungroupthem;

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