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首頁 > 研究所、轉學考(插大)◆工程數學 > 108年 - 108 國立臺灣大學_碩士班招生考試：工程數學(D)#124907 >

題組內容

6. X and Y are two random variables with a joint probability density function as

(a) Derive the conditional cumulative distribution function FXY(x|y). (5%)

其他申論題

(b) What is the mathematical definition of a probability function? (3%)

(c) Consider the experiment of throwing a fair 2-sided dice, where the sample space S = {?, ?} and Prob({?}) = Prob({?}) = 0.5. Define a random variable X for this experiment and write down the probability mass function PMF of X, i.e., Prob(X=x) for all real x ∈ R. (5%)

(a) You know that the previous taxi arrival is s time ago. Let TR be the remaining time you have to wait for the next taxi arrival. Derive the probability that you will need to wait for no more than a time duration r, knowing that the previous arrival was s time ago. (You need to give the derivation, not just the answer. 10%)

(b) Now (time 0), you decide not to take a taxi but to count the number of taxi arrivals in a period [0, τ]. Let the number of arrivals in [0, τ] be K, which is a RV. Derive Prob(K = 0). (You need to give the derivation, not just the answer. 5%)

(b) E[E[X|Y]] = ? (5%)

(c) Is the correlation coefficient ρXY = 0? Explain why. (5%)

(a) How do you estimate the value of θ? (5%)

(b) Is your estimate unbiased, i.e., does the mean of your estimate equal θ? Why? (4%)

1. 電腦化適性測驗與電腦化多階段測驗(computerized multi-stage test)

2. 測驗等化(test equating)與試題連結(item linking)

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