Let f (x) be the probability density function, f (x) = K(8 − x) , x = −1, − 2,
0, 1, 2 and f (x) = 0 , otherwise. Find the value of K.
If P(A) = 0.4 , P(B|A) = 0.3 , P(A ∪ B) = 0.6 , then P(B) = ?
Let X be a random variable with E[X] = 2 , E[X2 ] = 8 . Estimate
P(−3 < X < 7) by Chebyshev’s theorem.
Let f (x) be the probability density function of random variable X.
f (x) = ax3 , 0 < x < 1. In addition, P(X > b) = 0.05 . Calculate the value of
a and b.
Let X ~ N(μ ,σ 2 ) and Y = exp(X) . Express the probability density
function of Y.
We want to test whether a bath soap production process is meeting the
standard output of 110 bears per batch. Use a 0.05 level of significance for
the test and a planning value of 5 for the standard deviation. If the mean
output drops to 107 bears per batch, the firm wants to have a 90% chance
of conducting that the standard production output is not being met. How
large a sample should be selected?
Shorney Construction Company bids on projects assuming that the mean
idle time per worker is 70 or fewer minutes per day. A sample of 36
construction workers will be used to test this assumption under a 0.05
level of significance. Assume that the population standard deviation is 18
minutes. What is the probability of making a Type II error when the
population mean of idle time is 76 minutes?
Suppose a factory has two machines A and B that make 70% and 30% of
the total production, respectively. Machine A produces 5% defective items
of its output, while machine B produces 10% defective items of its output.
Find the probability that a given defective item was produced by machine
B.
Describe the formula of Bayes’ Theorem.
請以數學定義並且詳細說明,最佳的點估計值需符合那4 個特性。(12%)
(每一小題6%,共18%)請導出下列分配的動差母函數(請列出推導過程,無過
程不給分):
(1) 常態分配(平均數μ ,變異數σ 2)。
(2) 二項分配(試驗次數n,成功機率p)。
(3) 指數分配(平均數θ )。
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