課程名稱︰機率
課程性質︰資工系必修
課程教師︰陳文進
開課學院：電資學院
開課系所︰資工系
考試日期（年月日）︰2007/6/20
考試時限（分鐘）：3hrs
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
(1)(10%) Let X, Y, Z be three random variables with E(X) = E(Y) = 1, E(Z) = -1,
Var(X) = Var(Y) = Var(Z) = 1, and correlated coefficients ρ_xy = 0,
ρ_xz = 1/2, ρ_yz = -1/2. Find E(X + Y + Z) and Var(X + Y + Z).
(2)(15%) At a certain bank, the amount of time that a customer spends being
served by a teller is an exponential random variable with mean λ = 5
minutes. If there is a customer in service when you enter the bank, what is
the probability that the customer will still be with the teller after an
additional 4 minutes?
(3)(15%) If log_2 X ~ N(1, 4), find the probability P(1/2 < X < 2).
(4)(15%) If X ~ U(0, 1), find the pdf of the random variable Y = -2lnX.
(5)(15%) Let X, Y be two independent standard normal variable. Find the pdf of
Z = √(X^2 + Y^2).
(6)(15%) If X1, X2, ... is a sequence of independent and idnetically
distributed random variables with μ = E(Xi) and σ^2 = Var(Xi) < ∞,
i = 1, 2, .... Prove that for any ε > 0,
n
lim P(| [(2/n(n + 1))*Σ k*Xk] - μ | < ε) = 1.
n->∞ k = 1
(7)(15%) If the probability that a new born baby is a boy is 0.5. Use central
limiting theorem to find the probability that there are at least 4975 girls
among the 10000 new born babies.
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