题目
Problem
Use suitable approximating distributions to answer the following.
(a) The random variable W W W B ( 700 , 0.005 ) \mathrm{B}(700,0.005) B ( 700 , 0.005 )
(i) Find P ( W ≥ 4 ) \mathrm{P}(W\ge 4) P ( W ≥ 4 )
(3)
Two values of W W W
(ii) Find the probability that the sum of these two values is less than 3 3 3
(3)
(b) The random variable X X X P o ( 200 ) \mathrm{Po}(200) Po ( 200 )
Use a suitable approximating distribution to find P ( X > 205 ) \mathrm{P}(X>205) P ( X > 205 )
(4)
题目中文翻译
用合适的近似分布回答下列问题。
(a) 随机变量 W W W B ( 700 , 0.005 ) \mathrm{B}(700,0.005) B ( 700 , 0.005 )
(i) 求 P ( W ≥ 4 ) \mathrm{P}(W\ge 4) P ( W ≥ 4 )
(3)
随机选取 W W W
(ii) 求这两个取值之和小于 3 3 3
(3)
(b) 随机变量 X X X P o ( 200 ) \mathrm{Po}(200) Po ( 200 )
用一个合适的近似分布求 P ( X > 205 ) \mathrm{P}(X>205) P ( X > 205 )
(4)
原卷通用作答要求
Answer all questions. You should use a calculator where appropriate.
You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator.
Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.
请回答所有 题目。适当时应使用计算器。必须清楚展示所有必要的计算过程；仅给出计算器答案而没有过程，不予评分。除非题目另有精度要求，非精确数值答案保留 3 位有效数字，以度为单位的角度保留 1 位小数。
解答
(a)(i)
解法一
思路
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题目要求使用近似分布；二项试验次数大且成功概率小，取参数 3.5 3.5 3.5
答题过程
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Since n = 700 n=700 n = 700 p = 0.005 p=0.005 p = 0.005
W ≈ P o ( 700 ( 0.005 ) ) = P o ( 3.5 ) . W\approx\mathrm{Po}(700(0.005))=\mathrm{Po}(3.5). W ≈ Po ( 700 ( 0.005 )) = Po ( 3.5 ) . Then
P ( W ≥ 4 ) ≈ 1 − e − 3.5 ( 1 + 3.5 ) − e − 3.5 ( 3.5 2 2 ! + 3.5 3 3 ! ) ≈ 0.463 . \begin{align*}
\mathrm P(W\ge4)
\approx&\,1-\mathrm e^{-3.5}(1+3.5)\\[4mm]
&\,-\mathrm e^{-3.5}\bigg(\frac{3.5^2}{2!}+\frac{3.5^3}{3!}\bigg)\\[4mm]
\approx&\,\boxed{0.463}.
\end{align*} P ( W ≥ 4 ) ≈ ≈ 1 − e − 3.5 ( 1 + 3.5 ) − e − 3.5 ( 2 ! 3. 5 2 + 3 ! 3. 5 3 ) 0.463 .
(a)(ii)
解法一
思路
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两个独立观测值的近似 Poisson 参数相加，小于三个只包含零、一、二。
答题过程
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Let W 1 , W 2 W_1,W_2 W 1 , W 2 P o ( 7 ) \mathrm{Po}(7) Po ( 7 )
P ( W 1 + W 2 < 3 ) ≈ e − 7 ( 1 + 7 + 7 2 2 ! ) ≈ 0.0296 . \begin{align*}
&\,\mathrm P(W_1+W_2<3)\\[4mm]
\approx&\,\mathrm e^{-7}\bigg(1+7+\frac{7^2}{2!}\bigg)\\[4mm]
\approx&\,\boxed{0.0296}.
\end{align*} ≈ ≈ P ( W 1 + W 2 < 3 ) e − 7 ( 1 + 7 + 2 ! 7 2 ) 0.0296 .
解法二
思路
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也可在两个 Poisson 近似下列出总数为零、一、二的全部六种互斥组合，再使用独立性相乘。
答题过程
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Approximate W 1 , W 2 W_1,W_2 W 1 , W 2 P o ( 3.5 ) \mathrm{Po}(3.5) Po ( 3.5 ) 3 3 3
( 0 , 0 ) , ( 1 , 0 ) , ( 0 , 1 ) , ( 2 , 0 ) , ( 1 , 1 ) , ( 0 , 2 ) . \begin{align*}
&\,(0,0),\ (1,0),\ (0,1),\\[4mm]
&\,(2,0),\ (1,1),\ (0,2).
\end{align*} ( 0 , 0 ) , ( 1 , 0 ) , ( 0 , 1 ) , ( 2 , 0 ) , ( 1 , 1 ) , ( 0 , 2 ) . Their probabilities sum to
P ( W 1 + W 2 < 3 ) ≈ e − 7 ( 1 + 3.5 + 3.5 ) + e − 7 ( 3.5 2 2 ! + 3.5 2 + 3.5 2 2 ! ) = e − 7 ( 1 + 7 + 24.5 ) ≈ 0.0296 . \begin{align*}
&\,\mathrm P(W_1+W_2<3)\\[4mm]
\approx&\,\mathrm e^{-7}(1+3.5+3.5)\\[4mm]
&\,+\mathrm e^{-7}\bigg(\frac{3.5^2}{2!}+3.5^2+\frac{3.5^2}{2!}\bigg)\\[4mm]
=&\,\mathrm e^{-7}(1+7+24.5)\\[4mm]
\approx&\,\boxed{0.0296}.
\end{align*} ≈ = ≈ P ( W 1 + W 2 < 3 ) e − 7 ( 1 + 3.5 + 3.5 ) + e − 7 ( 2 ! 3. 5 2 + 3. 5 2 + 2 ! 3. 5 2 ) e − 7 ( 1 + 7 + 24.5 ) 0.0296 .
(b)
解法一
思路
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Poisson 参数 200 200 200 205.5 205.5 205.5
答题过程
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Since X ∼ P o ( 200 ) X\sim\mathrm{Po}(200) X ∼ Po ( 200 ) 200 > 15 200>15 200 > 15 X X X Y ∼ N ( 200 , 200 ) Y\sim\mathrm N(200,200) Y ∼ N ( 200 , 200 )
P ( X > 205 ) = P ( X ≥ 206 ) ≈ P ( Y > 205.5 ) = 1 − Φ ( 205.5 − 200 200 ) = 1 − Φ ( 0.388909 … ) ≈ 0.349 . \begin{align*}
\mathrm P(X>205)
=&\,\mathrm P(X\ge206)\\[4mm]
\approx&\,\mathrm P(Y>205.5)\\[4mm]
=&\,1-\Phi\bigg(\frac{205.5-200}{\sqrt{200}}\bigg)\\[4mm]
=&\,1-\Phi(0.388909\ldots)\\[4mm]
\approx&\,\boxed{0.349}.
\end{align*} P ( X > 205 ) = ≈ = = ≈ P ( X ≥ 206 ) P ( Y > 205.5 ) 1 − Φ ( 200 205.5 − 200 ) 1 − Φ ( 0.388909 … ) 0.349 . 