(2) 若你改變心意，想加碼投資到 \( 10萬元，則你可能的淨回收(net payoff)的平均數 \) \( = \)  __________，與變異數 \( = \) __________

(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.

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