課程名稱︰機率導論 課程性質︰數學系必修 課程教師︰陳鵬文 開課學院：理學院 開課系所︰數學系 考試日期（年月日）︰2010年04月21日 考試時限（分鐘）：110分鐘 是否需發放獎勵金：是 試題 : 1. For each part, if the statement is true, circle the printed capital T. If statement is false, circle the printed capital F. (a) Suppose that the life distribution of an item has the hazard rate function, λ(t) = t^2, t > 0. Let Ｘ be the lifetime of the item. Then Ｘ is memoryless. T F [3] (b) If Ｘ is a negative binomial random variable with parameters (r,p) = ( 2 , 1/5 ), then E[Ｘ] = 2/5. T F [3] (c) If Ｘ,Ｙ are random variables with Var(Ｘ) = 1, Var(Ｙ) = 2, then Var(Ｘ+Ｙ) = 3. T F [3] (d) If Ｘ,Ｙ are Poisson random variables with E[Ｘ] > E[Ｙ], then Var(Ｘ) > Var(Ｙ). T F [3] (e) Let Ａ,Ｂ and Ｃ be events relating to the experiment of rolling a pair of dice. If P(Ａ|Ｃ) > P(Ｂ|Ｃ) and P(Ａ|Ｃ^c) > P(Ｂ|Ｃ^c), then P(Ａ) > P(Ｂ). T F [3] 2. The expected number of typographical errors on a page of a certain magazine is 1. What is the probability that the next page you read contains 2 or more typographical errors. [10] 3. (a) Let Ｘ be a normal random variable with mean 2 and variance 4. Find the value of c such that P{ 1 < Ｘ < c } = 0.5. [5] (b) Find E[ (1+2Ｘ)^2 ]. [5] 4. Consider n married couples randomly seated in a round table. (a) If n = 4, find the exact probability that no husband sits next to his wife. [5] (b) Consider the case that n tends to ∞. Use the Poisson approximation to estimate the probability that no husband sits next to his wife. [5] 5. People enter a gambling casino at a rate of 1 every 2 minutes. (a) What is the probability that at least 3 people enter the casino between 12:00 and 12:05. [5] (b) Find the probability distribution of the time, starting from now, until the next person entering the casino. [5] 6. Die Ａ has 4 red and 2 white faces, whereas die Ｂ has 3 red and 3 white faces. A fair coin is flipped once. If it lands on heads, the game continues with die Ａ; if it lands on tails, then die Ｂ is to be used.If the first throw results in red, what is the probability of red at the second throw? [10] 7. The probability of getting a head on a single toss of a coin is 1/3. Suppose that Ａ starts and continues to flip the coin until a tail shows up, at which point Ｂ starts flipping. Then Ｂ continues to flip until a tail comes up, at which point Ａ takes over. Let Ｐ_n,m denote the probability that the person flipping the coin accumulates a total of n heads before the other person accumulates m heads (You should derive the relation between Ｐ_n,m and Ｐ_n-1,m first). (a) Let Ｅ be the desired event. Let Ｈ be the event that the first trial results in a head. By conditioning on the outcomes of the first trail, establish an equation for Ｐ_n,m . [3] (b) Find Ｐ_0,1 , Ｐ_1,0 and Ｐ_1,1 . [7] 8. Let Ｘ be the number of times that a fair coin that is flipped 100 times lands on heads. Use the normal approximation to find the probability that Ｘ < 41. [12] 9. Let Ｘ be uniformly distributed over (0,2). Let Ｙ = 1 / (Ｘ+1). (a) Find the probability density function of the random variable Ｙ. [5] (b) Find E[Ｙ] and Var(Ｙ). [8] --

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