連續(xù)型概率分布

連續(xù)型概率分布均勻概率分布均勻概率分布：一種連續(xù)型概率分布，其隨機(jī)變量在等長度的每一區(qū)間上取值的概率都相同。例：隨機(jī)變量x

代表一個航班從芝加哥飛往紐約的飛行時間。假定飛行時間可以是從120分鐘到140分鐘區(qū)間內(nèi)的任意值，在120-140分鐘的區(qū)間內(nèi)飛行時間在任意1分鐘區(qū)間內(nèi)的概率都相同。隨機(jī)變量x

就說是有均勻概率分布。第2頁,共25頁，2024年2月25日，星期天均勻概率密度函數(shù)均勻概率密度函數(shù)在連續(xù)情況下，概率函數(shù)的對應(yīng)物是概率密度函數(shù)，也用f(x)表示,

f(x)給出x

的任一特定值的概率密度；它不直接給出隨機(jī)變量取某一特定值的概率。第3頁,共25頁，2024年2月25日，星期天作為概率度量的面積f(x)0.050第4頁,共25頁，2024年2月25日，星期天連續(xù)型隨機(jī)變量和離散隨機(jī)變量的區(qū)別不再討論隨機(jī)變量取某一特定值的概率。代替地，討論隨機(jī)變量在某一給定區(qū)間取值的概率。隨機(jī)變量在從x1到x2間的某一給定區(qū)間取值的概率被定義為概率密度函數(shù)在x1與x2間的圖形的面積。第5頁,共25頁，2024年2月25日，星期天均勻連續(xù)型概率分布的期望值和方差第6頁,共25頁，2024年2月25日，星期天練習(xí)已知隨機(jī)變量x在10和20間服從均勻分布計算P(x<15)；計算P(12≤x≤18)；計算E(x)；計算Var(x)。第7頁,共25頁，2024年2月25日，星期天6連續(xù)型概率分布均勻概率分布正態(tài)概率分布二項(xiàng)概率的正態(tài)近似指數(shù)概率分布第8頁,共25頁，2024年2月25日，星期天正態(tài)曲線正態(tài)概率密度函數(shù)xf(x)第9頁,共25頁，2024年2月25日，星期天正態(tài)概率分布的性質(zhì)正態(tài)概率分布有一個完整家族。每一特定正態(tài)分布通過其均值μ

、標(biāo)準(zhǔn)差σ

來區(qū)分。正態(tài)曲線的最高點(diǎn)在均值，它也是分布的中位數(shù)和眾數(shù)分布的均值可以是任意數(shù)值：負(fù)數(shù)、零或正數(shù)。正態(tài)概率分布是對稱的。曲線的尾端向兩個方向無限延伸，且理論上永遠(yuǎn)不會與橫軸相交。第10頁,共25頁，2024年2月25日，星期天正態(tài)概率分布的性質(zhì)標(biāo)準(zhǔn)差決定曲線的寬度正態(tài)概率分布曲線下的總面積是1，對所有的連續(xù)型概率分布都是如此。正態(tài)隨機(jī)變量的概率由曲線下面積給出。一些常用區(qū)間的概率是68.26%，95.44%，99.72%第11頁,共25頁，2024年2月25日，星期天標(biāo)準(zhǔn)正態(tài)概率分布均值為0、標(biāo)準(zhǔn)差為1的正態(tài)分布。

=1Z標(biāo)準(zhǔn)正態(tài)分布

第12頁,共25頁，2024年2月25日，星期天z0.000.010.020.030.040.050.060.070.080.090.00.50000.50400.50800.51200.51600.51990.52390.52790.53190.53590.10.53980.54380.54780.55170.55570.55960.56360.56750.57140.57530.20.57930.58320.58710.59100.59480.59870.60260.60640.61030.61410.30.61790.62170.62550.62930.63310.63680.64060.64430.64800.65170.40.65540.65910.66280.66640.67000.67360.67720.68080.68440.68790.50.69150.69500.69850.70190.70540.70880.71230.71570.71900.72240.60.72570.72910.73240.73570.73890.74220.74540.74860.75170.75490.70.75800.76110.76420.76730.77040.77340.77640.77940.78230.78520.80.78810.79100.79390.79670.79950.80230.80510.80780.81060.81330.90.81590.81860.82120.82380.82640.82890.83150.83400.83650.83891.00.84130.84380.84610.84850.85080.85310.85540.85770.85990.86211.10.86430.86650.86860.87080.87290.87490.87700.87900.88100.88301.20.88490.88690.88880.89070.89250.89440.89620.89800.89970.90151.30.90320.90490.90660.90820.90990.91150.91310.91470.91620.91771.40.91920.92070.92220.92360.92510.92650.92790.92920.93060.93191.50.93320.93450.93570.93700.93820.93940.94060.94180.94290.94411.60.94520.94630.94740.94840.94950.95050.95150.95250.95350.95451.70.95540.95640.95730.95820.95910.95990.96080.96160.96250.96331.80.96410.96490.96560.96640.96710.96780.96860.96930.96990.97061.90.97130.97190.97260.97320.97380.97440.97500.97560.97610.97672.00.97720.97780.97830.97880.97930.97980.98030.98080.98120.98172.10.98210.98260.98300.98340.98380.98420.98460.98500.98540.98572.20.98610.98640.98680.98710.98750.98780.98810.98840.98870.98902.30.98930.98960.98980.99010.99040.99060.99090.99110.99130.99162.40.99180.99200.99220.99250.99270.99290.99310.99320.99340.99362.50.99380.99400.99410.99430.99450.99460.99480.99490.99510.99522.60.99530.99550.99560.99570.99590.99600.99610.99620.99630.99642.70.99650.99660.99670.99680.99690.99700.99710.99720.99730.99742.80.99740.99750.99760.99770.99770.99780.99790.99790.99800.99812.90.99810.99820.99820.99830.99840.99840.99850.99850.99860.99863.00.99870.99870.99870.99880.99880.99890.99890.99890.99900.9990第13頁,共25頁，2024年2月25日，星期天計算任一正態(tài)概率分布的概率標(biāo)準(zhǔn)正態(tài)分布變換第14頁,共25頁，2024年2月25日，星期天例：Grear輪胎公司問題Grear輪胎公司剛剛開發(fā)了一種新的鋼帶子午線輪胎，根據(jù)對輪胎的實(shí)際道路測試，工程小組已估計出輪胎可行駛里程的均值=36

500英里，標(biāo)準(zhǔn)差=5

000。另外，收集的數(shù)據(jù)表明正態(tài)分布是一合理的假設(shè)。輪胎預(yù)期使用超過40000英里的百分率是多少？假設(shè)Grear公司正在考慮一項(xiàng)擔(dān)保：如果初始的輪胎沒有超過擔(dān)保中設(shè)定的里程，公司將折價更換輪胎。如果Grear公司希望符合折價擔(dān)保的輪胎不超過10%，則擔(dān)保里程應(yīng)為多少？第15頁,共25頁，2024年2月25日，星期天z0.000.010.020.030.040.050.060.070.080.090.00.50000.50400.50800.51200.51600.51990.52390.52790.53190.53590.10.53980.54380.54780.55170.55570.55960.56360.56750.57140.57530.20.57930.58320.58710.59100.59480.59870.60260.60640.61030.61410.30.61790.62170.62550.62930.63310.63680.64060.64430.64800.65170.40.65540.65910.66280.66640.67000.67360.67720.68080.68440.68790.50.69150.69500.69850.70190.70540.70880.71230.71570.71900.72240.60.72570.72910.73240.73570.73890.74220.74540.74860.75170.75490.70.75800.76110.76420.76730.77040.77340.77640.77940.78230.78520.80.78810.79100.79390.79670.79950.80230.80510.80780.81060.81330.90.81590.81860.82120.82380.82640.82890.83150.83400.83650.83891.00.84130.84380.84610.84850.85080.85310.85540.85770.85990.86211.10.86430.86650.86860.87080.87290.87490.87700.87900.88100.88301.20.88490.88690.88880.89070.89250.89440.89620.89800.89970.90151.30.90320.90490.90660.90820.90990.91150.91310.91470.91620.91771.40.91920.92070.92220.92360.92510.92650.92790.92920.93060.93191.50.93320.93450.93570.93700.93820.93940.94060.94180.94290.94411.60.94520.94630.94740.94840.94950.95050.95150.95250.95350.95451.70.95540.95640.95730.95820.95910.95990.96080.96160.96250.96331.80.96410.96490.96560.96640.96710.96780.96860.96930.96990.97061.90.97130.97190.97260.97320.97380.97440.97500.97560.97610.97672.00.97720.97780.97830.97880.97930.97980.98030.98080.98120.98172.10.98210.98260.98300.98340.98380.98420.98460.98500.98540.98572.20.98610.98640.98680.98710.98750.98780.98810.98840.98870.98902.30.98930.98960.98980.99010.99040.99060.99090.99110.99130.99162.40.99180.99200.99220.99250.99270.99290.99310.99320.99340.99362.50.99380.99400.99410.99430.99450.99460.99480.99490.99510.99522.60.99530.99550.99560.99570.99590.99600.99610.99620.99630.99642.70.99650.99660.99670.99680.99690.99700.99710.99720.99730.99742.80.99740.99750.99760.99770.99770.99780.99790.99790.99800.99812.90.99810.99820.99820.99830.99840.99840.99850.99850.99860.99863.00.99870.99870.99870.99880.99880.99890.99890.99890.99900.9990第16頁,共25頁，2024年2月25日，星期天練習(xí)紐約證券交易所的交易量近年來已增長。1998年1月前2周的平均日交易量是646百萬股。每日成交量的概率分布近似正態(tài)，標(biāo)準(zhǔn)差大約100百萬股。a.交易量低于400百萬股的概率是多少？b.交易量超過800百萬股的時間百分率是多少？c.如果交易所打算在成交量最高的5%的交易日發(fā)布新聞，引發(fā)新聞的成交量是多大？第17頁,共25頁，2024年2月25日，星期天6連續(xù)型概率分布均勻概率分布正態(tài)概率分布二項(xiàng)概率的正態(tài)近似指數(shù)概率分布第18頁,共25頁，2024年2月25日，星期天二項(xiàng)概率的正態(tài)近似在試驗(yàn)數(shù)大于20，np≥5和n(1-p)≥5情況下，正態(tài)概率分布給出一易于使用的二項(xiàng)概率近似。例：某公司有10%的發(fā)票出錯的歷史。一個有100張發(fā)票的樣本已選好，計算有13張發(fā)票有錯的概率。連續(xù)性校正因子：當(dāng)用連續(xù)正態(tài)概率分布來近似離散二項(xiàng)概率分布時，從x值加減的0.5值。第19頁,共25頁，2024年2月25日，星期天練習(xí)已知一種主要的全國信用卡的所有持卡人中有30%在發(fā)生任何利息之前已全額支付了其帳單。利用二項(xiàng)分布的正態(tài)近似對一個有150名信用卡持卡人的組回答下列問題：a.在任何利息費(fèi)用發(fā)生之前已支付其帳款余額的客戶數(shù)在40到60之間的概率是多大？b.在任意利息費(fèi)用發(fā)生之前已支付其帳款余額的客戶數(shù)為30或更少的概率是多少？第20頁,共25頁，2024年2月25日，星期天z0.000.010.020.030.040.050.060.070.080.090.00.50000.50400.50800.51200.51600.51990.52390.52790.53190.53590.10.53980.54380.54780.55170.55570.55960.56360.56750.57140.57530.20.57930.58320.58710.59100.59480.59870.60260.60640.61030.61410.30.61790.62170.62550.62930.63310.63680.64060.64430.64800.65170.40.65540.65910.66280.66640.67000.67360.67720.68080.68440.68790.50.69150.69500.69850.70190.70540.70880.71230.71570.71900.72240.60.72570.72910.73240.73570.73890.74220.74540.74860.75170.75490.70.75800.76110.76420.76730.77040.77340.77640.77940.78230.78520.80.78810.79100.79390.79670.79950.80230.80510.80780.81060.81330.90.81590.81860.82120.82380.82640.82890.83150.83400.83650.83891.00.84130.84380.84610.84850.85080.85310.85540.85770.85990.86211.10.86430.86650.86860.87080.87290.87490.87700.87900.88100.88301.20.88490.88690.88880.89070.89250.89440.89620.89800.89970.90151.30.90320.90490.90660.90820.90990.91150.91310.91470.91620.91771.40.91920.92070.92220.92360.92510.92650.92790.92920.93060.93191.50.93320.93450.93570.93700.93820.93940.94060.94180.94290.94411.60.94520.94630.94740.94840.94950.95050.95150.95250.95350.95451.70.95540.95640.95730.95820.95910.95990.96080.96160.96250.96331.80.96410.96490.96560.96640.96710.96780.96860.96930.96990.97061.90.97130.97190.97260.97320.97380.97440.97500.97560.97610.97672.00.97720.97780.97830.97880.97930.97980.98030.98080.98120.98172.10.98210.98260.98300.98340.98380.98420.98460.98500.98540.98572.20.98610.98640.98680.9