(1) 隨機變數 \( X \) 的平均數 \( = \)  __________及變異數 \( = \) __________；

(2) \( \mathrm{P}(\mathrm{X} = 0) = \)  __________，和 \( \mathrm{P}(\mathrm{X}\geq 2) = \) __________

(1) 此組樣本的變異數大於 18.69 公克的機率 =_____________；

(2) \( P(5.5728 \leq S^2 \leq 22.063) = ? \)__________；

1. There is one monopolist that utilizes one input, say labor, \( L \), which he acquires at the fixed cost (per unit is $8) to produce the output, \( Q \). In addition, both demand and production functions are \( p = 60 - 2q \) and \( q = 4\sqrt{L} \), respectively. If profit maximization for this monopolist is proven achievable, please solve the values of \( p, q \), and \( L \). (25%)

2. Assuming a competitive market in the U.S., together with a supply function: \( q{s} = 400p - 400 \) and a demand function: \( q{d} = 2400 - 400p \). When the equilibrium price and quantity are reached, what are equilibrium price and quantity? (10%). Now due to Trump's reciprocal tariff policy, the authorities impose a $4 per-unit tax to be collected from sellers. We denote \( P{c} \) as the price paid by consumers after imposed tax and \( P{s} \) as the price received by producers after tax. Please solve the after-tax equilibrium quantity (\( q^{} \)) and equilibrium prices for both consumers \( P{c}^{} \) and sellers \( P{s}^{*} \). (15%)

(a) flexible exchange rate regime. (10%)

(b) fixed exchange rate regime. (10%)

(a) Please mathematically express and explain interest parity \( (5\%) \) and covered interest parity, \( (5\%) \) respectively.

(b) Please mathematically express and explain the Fisher effect describing the relationship between nominal interest rates and inflation across two countries i and j. (5%) Please mathematically prove the Fisher effect (5%). Please note that you are allowed to make necessary assumptions for the proof.

3. If the central bank appears concerned about the domestic economy overheating due to strong demand of the exports. Will preventing the domestic currency from appreciating help resolve this issue? Why or why not? Please graphically explain and justify your answer. (10%)

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