90年 - 90 國立交通大學管碩士班考試入學試題_交通運輸研究所、運輸科技與管理學系：統計學#124782

(1). two random variables are independent and identically distributed. (6%)

(2). Central Limit Theorem. (6%)

2. Given the joint probability distribution of two random variables X and Y as following, (4%)

3. 普松隨機變數(Poisson random variable)X之機率函數為
P(x=k) = , 試証其符合機率總和等於1的特性。(8%)

 (1).對一個隨機變數X而言,其平均值(mean)與變異數(variance)之意
義各為何? (6%)

(2). 試証 Var(X) = E(X²) - [E(X)]² (8%)

5. 一隨機變數X之機率密度函數(probability density function)為

試繪製此隨機變數之機率密度函數與累積機率函數(cumulative distribution function)的圖形。(6%)

6. The manager of a convenient store wants to know whether the customer arrival in every ten-minute period follows Poisson distribution. The alternatives are as follows:
H₀: The population distribution is Poisson.
H₁: The population distribution is not Poisson.
One day he observes from 8 a.m. to 4 p.m. The data on the number of customers (X) arriving at his store in each ten-minute period are as follows:

Hence the sample mean = 6.0.
Using the Chi-square procedure, for α = 0.05, test whether the probability distribution of the ten-minute customer arrival is Poisson. (20%)

7. Consider the simple linear regression model
Yᵢ = β₀ + β₁Xᵢ + εᵢ, Vᵢ = 1,2,...,n, and εᵢ's are i.i.d. N(0, σ²),
Show that the least squares estimators of parameters β₀ and β₁ satisfy
b₁ = ， and
b₀ =
whereare the sample means of the Xᵢ and Yᵢ observations,
respectively. (12%)

(1). Construct a scatter plot of the data. (5%)

(2). Calculate b₀ and b₁. Obtain the (least squares) estimated regression function and plot it on the graph in part (1). If the monthly income of Tony's is NT$ 60,000, then how much electricity fee might he pay last month? (10%)

(3). Interpret b₀ and b₁. (5%)