[問題] hogg習題
是 probability and statistical inference 8e 的 3.3-23
An insusurance agent receives a bonus if the loss ratio L on his business
is less than 0.5, where L is the total losses (say, X) divided by the
total premiums (say, T). The bonus equals (0.5-L)(T/30) if L < 0.5 and
equals zero otherwise. If X (in $100,000) has the p.d.f
f(x)= 3/x^4, x > 1,
and if T (in $100,000) equals 3, determine the expected value of the bonus.
我的想法是：
要求 E((0.5-L)(T/30))的值，就必須得到 E(L)
又 X/T = X/3 = L ，所以 E(X/3) = (1/3)E(X) = E(L)
1.5
E(X)=∫ x．3/x^4 dx = 5/6 E(L) = 5/18
1
代入原式 E( (0.5-5/18)(3/30) ) = 0.022
所以bonus就是 0.0222 x $100,000 = $2222.22
然後關於求E(X)積分的上限問題，我試過用無限大代入，不過這樣E(L)就變成0.5
所以我就想說 L < 0.5 表示 X < 1.5 ，所以才用1.5
不過這題答案是 $740.74
麻煩請指正我一下，謝謝！！
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