武汉理工大学概率论与数理统计英文版试题

with confidence
coefficient1− α. . 区间估计
1
得分
II. (10′) In a city, 50.2 percent of the people are men and 49.8 percent of the
people are women. Records show that the probability that a man has a certain disease is 0.05 and the
person is found to have the disease. 全概率公式及贝叶斯公式 要求：1.必须要设事件
2.将题目所给的数据用概率的形式给出来 3.要有公式 详细过程见 exercise 3，习题课里每个题目都写了详细的过程
得分
III. (10′ ).Suppose that the p.d.f. of a random variable X is as follows:
写 考
random sample showed an average of 23500km and a standard deviation of 3900km?
生 信
Use a 0.01 level of significance.
息
单个正态总体均值的假设检验，详见 exercise 15，参考答案已经上传在参考资源里
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is the unbiased estimator .of λ−1 . Then α =
； 无偏估计的概念
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10. Suppose that X1, X2, …, Xn are a random sample from X~ N (μ,σ 2 ) and μ,σ 2 are
…
…
unknown. The confidence interval for μ is
卷 装
+ ( X 4 + X 5 + X 6 )2 and cY has a χ 2 distribution. Then c =
；抽样分布
订
9. Suppose that X1, X 2 , X 3, X 4 is a random sample from X ~ E(λ) and T =α(X1 + X2 + X3 + X4)
不 要
than 20000km per year. To test this claim, a random sample of 100 automobile owners 填
are asked to keep a record of the km they travel. Would you agree this claim if the
components now, what is the probability that the total length of life of these components
is more than 2000 hours? (Use the Central Limit Theorems)
利用中心极限定理计算，详见 exercise 12，参考答案已经上传在参考资源里
装
1. Put 3 balls into 4 boxes randomly. The probability of that there is at most one ball in each
订 线
box is
；古典概率的计算
内 不
2. Suppose A and B are independent, and P(A) = 0.6 and P(A+B) = 0.8. Then P(B A ) = ；
3
得分
VII(10′ )Suppose that X has the following distribution：
X
1
2
3
p
θ2
2θ (1 − θ )
(1 − θ )2
… …
where the parameterθ (0 < θ < 1) is unknown. Given the observed value x1 = 1, x2 = 2,
X -1 0
1
Y
0
1
p 1/4 1/2 1/4
p 1/2 1/2
and P{XY = 0} = 1.a) Find the joint distribution of (X ,Y ) ; b) Determine whether or not X
and Y are independent.
二维离散型随机变量的联合分布、边缘分布、条件分布和独立性的判断
要求：1.要设变量 2.用中心极限定理是近似计算，要用 ≈
得分
V. (10′ ) Suppose X ~ E(2), Determine the p.d.f. of Y = 1 − e−2X .
随机变量函数的分布
这部分内容上过习题课，exercise 7 再次强调：一定要用分布函数法求解
得分
VI. (10′ ) Given that X and Y have the following distribution：
… …
x3 = 1, determine the moment estimators and the maximum likelihood estimator ofθ .
试
卷
装
订
点估计，详见 exercise 14，参考答案已经上传在参考资源里
线
… … … … … …
装
订
线
内
不
要
答
题ห้องสมุดไป่ตู้
，
得分
VIII. (10′ )It is claimed that an automobile is driven on the average less
Φ(0) = 0.5, Φ(1) = 0.8413, Φ(1.645) = 0.95, Φ(1.96) = 0.975, Φ(2) = 0.9772,
…
Φ(2.327) = 0.99, Φ(2.567) = 0.995, 试卷所需的查表数据
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专业
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得分
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班级
I. Fill in the blanks ( 3′ ×10).
2
得分
IV.(10′ ) Suppose that the length of life of a certain electrical component
has an exponential distribution with 100hour as its mean. Randomly taking 16 such
要 填
then μ =
；一维离散型和连续型随机变量概率分布，常见分布
写 考 生 信
5. Suppose ( X ,Y ) has
f
(x,
y)
=
⎧2e ⎨
−(2x+
y)
,
x > 0, y > 0 ，then P{Y ≤ X }=
⎩ 0 , otherwise
；二维
息
…
6. Suppose that X~N(0, 1),Y ~U(0, 1), and A and B are independent. Then D(X-2Y+4) = ；
武汉理工大学考试试卷（A 卷）
2010 ~2011 学年 1 学期 概率论与数理统计 课程 时间 120 分钟
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试
卷
学院
装
订
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56 学时， 3 学分，闭卷，总分 100 分，占总评成绩 80 % 2011 年 1 月 5 日 题号 一 二 三 四 五 六 七 八 九 十 合计
满分
100
得分
… …
… …
7. Suppose that X and Y have the same distribution and X ~ B(20, 0.1), ρ XY = 0.5 , then
… …
姓名
D(X + Y ) =
；数字特征的计算：常见分布的期望和方差，计算性质
… 试
8. Suppose that that X1, X2, …, X6 is a random sample from X ~ N (0,1). Y = (X1 + X2 + X3)2
要
由事件的关系和运算以及概率的性质等计算事件的概率
答 题
学 号 3. Suppose X has a Poisson distribution with λ = 1, then P{X = E( X )}=
；
， 不
4. Suppose X ~ N (μ,σ 2 ) and the probability of that y 2 + 4 y + X = 0 has no real root is 0.5.
f
(
x)
=
⎧ ⎪ ⎨
A
cos
x,
⎪ 0,
⎩
| x |≤ π 4
| x |> π 4
a)
Find the value of the constant A and
P⎨⎧0 ⎩
<
X
<
π 6
⎫ ⎬ ⎭
;
b)Find
the
distribution
function
F(x)
of
X
.
一维连续型随机变量的分布，详见 exercise 6 的 2、3 两题
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试
卷
装
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4
5
probability that a woman has the disease is 0.01. If a person in the city is selected at random, a) find
the probability that he has the disease. b) find the probability that the person is a woman given that the