期待値を定める前と期待値を定めた後の違い
期待値を定める前
平均は、起こり得る値の中心。
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的に向けてダーツを投げることを繰り返すと、その人の傾向(腕の良さと癖)が分かる。
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朝の通勤ラッシュの時間帯の電車は平均して、ダイヤから2,3分前後遅れる傾向にある。
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- サイコロの目を投げても、特にどの値が出易いという傾向はない。
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最頻値は、起きる確率が最も高い事象または要素。
- どの大学でも、入学した学生のうち、他の大学が第一志望だった者が一番多く、その次にこの大学が第一志望だった者が続く傾向にある。受験倍率が高い学校は特にこの傾向が強い。
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- 日本人の男性の平均身長は170cm前後、女性の平均身長は160cm前後であり、最頻値もそれぞれ、その付近になる。だからといって、男性で160cmの人や女性で170cmの人が珍しい訳でもない。
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期待値を定めた後
標本空間が等間隔で目盛が並ぶ数直線で、確率を数直線上のすべての点(目盛)に与える。
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この時、確率分布の平均は、確率関数で重みづけした標本の重心。
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標本空間が数直線(実数)の全体または一部で、確率を数直線上のすべての区間に与える。そのために確率密度関数を定める。
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この時、平均は、確率密度関数で重みづけした標本空間の重心。
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期待値の役割
確率関数、確率密度関数、そして期待値は、確率論の数学レベルを高めるために必要な演算といえる。