私立中原大學八十八學年度博士班招生考試命題紙

所組別：

工業工程學系博士班

科目：

工業工程基礎科目　　--工程統計

考試時間：

06月09日第 1節

工統 1. (10分)
Suppose that X and Y are independent continuous random variables having densities f_x and f_y , respectively.

(a)	Compute P{X < Y}.
(b)	Find the distribution of X+Y.

工統 2. (10分)
Using Moment Generating Functions , show that the following are true.

(a)	If X and Y are independent binomial random variables with parameters (n,p) and (m,p), respectively, then the distribution of X+Y is also binomial with (m+n,p).
(b)	If X and Y are independent normal random variables with parameters and , respectively, then X+Y is normal with .

工統 3. (15分)
Consider a regression model of tool wear (Y) on tool speed (X). In matrix terms, the regression equation is .

(a)	In matrix notation, the least squares method would require the minimization of .　Show that the least square solution for β would be in matrix operations.
(b)	What are the assumptions regarding ε ?
(c)	Suppose that anther independent variable “tool type”is also included in the model. There are four different tool types available.　Define a set of indicator (dummy) variables for tool type so that it can be included in the regression model.

工統 4. (15分)
Let y₁,y₂,...,y_n be a random sample of n observations from a normal distribution with mean μ and variance .　Show that the sample variance is an unbiased estimator of population variance when:

(a)	the sampled population has a normal distribution.
(b)	The distribution of the sampled population is unknown.

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