10. 可由同步發電機之無載飽和特性曲線及短路特性曲線求得，其值即為同一磁場電流下之電樞開路電壓與電樞短路電流之比值稱為__________ 。

1. Show that the mean of a hypergeometric random variable X, the number of successes in a random sample of size n selected from N items of which k are labeled success and N - k labeled failure, is nk/N.

2. The amount of lateral expansion (mils) was determined for a sample of n=9 pulsed-power gas metal arc welds used in LNG ship containment tanks. The resulting sample standard deviation was s=2.81 mils. Assuming normality, derive a 95% confidence intervals for σ².

a. Based on these 100 observations, is a Poisson distribution with a mean of 2.4 an appropriate model? Perform a goodness-of-fit test with α=0.05.

b. Perform the test using P-value approach.

a. Do the data suggest that the two methods provide the same mean value for natural vibration frequency? Using the P-value approach.

b. Find a 95% confidence interval on the mean difference between the two methods and use it to answer the question in part (a).

5. Let X be a random variable with distribution function   Plot the graph of F(x) and then evaluate E(x).

6. There are two classes of Probability and Statistics course. The problems of the final exam are identical for the two classes. Let μ₁ and μ₂ denote the population means of scores of class A and class B, respectively, and σ₁ and σ₂ denote the population variances of scores of class A and class B, respectively. A random sample of 13 students are taken from class A and 11 students are taken from class B. The sample average and standard deviation of scores of each class are class A: X̄ₐ = 84, Sₐ = 4 and class B: X̄բ = 77, Sբ = 6. Test whether the population means of scores of the two classes are equal. Make assumptions if necessary.

a. What is the probability function of Y?

b. What is the cumulative distribution function of Y?

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