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題組內容

4. The mathematical notion of random variable (RV) is very important in probability theory.

(a) What is the mathematical definition of a RV? (3%)

其他申論題

(c) (6%) Find , where e1 = .

(d) (2%) Find the nullity of A.

(a) (4%) Find T.

(b) (6%) Find T

(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%)

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

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

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