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課程名稱︰機率導論
課程性質︰數學系大二必修
課程教師︰陳宏
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2015/03/31
考試時限（分鐘）：60
試題 :
1. (20 points) The probability that n electrons are emitted during a certian
interval from a photoelectrically emissive surface under incident light is
n
P(n) = Cp ,
where n can be 0, 1, 2, ... and p is a positive number less than 1. Calculate
the constant C (it will depend on p). (Hint: you do not have to know anything
about Physics to solve this problem.)
2. (24 points) A random variable X has a cumulative distribution function given
by
╭ 0 x ＜ 0
│ 3 4
F (x) = ╯ 4x - 3x 0 ≦ x ＜ 1
X │
╰ 1 x ≧ 1
Find P(X = 1/2), P(X = 5), and P(X = 1/4).
3. (26 points) Alice and Bob are given copies of the same text for independent
proofreading. Alice finds 20 errors and Bob finds 15 errors, of which 10 were
found by Alice as well. Assume that Alice and Bob independently detect any
given error with (unknown) probabilities p and p , and that Alice and Bob
A B
each have detected expected numbers of errors. Estimate n which the total
number of errors in the text.
4. (25 points) Professor Whiskey has a class of n students, and he has to hand
back a quiz and a homework assignment. True to form, he hands back the 2n
papers haphazardly, uniformly randomly among all the possible ways to give
each student two items. (Note that he does not necessarily give each student
a quiz and a homework; some people might get two quizzes, others two
homeworks.) Let X be the number of students who end up with their own quiz
and homework. compute E(X) and Var(X).
Hint: You may want to let E be the event that student i gets both of her own
i
items back, and let X be the indicator of E . find P(E ) and P(E |E ) where
i i i j i
j ≠ i.
5. (25 points) If we try a certain experiment n times with a probability p of
success each time, show that the most likely number of successes is k = np.
(Assmue that p is a fraction and n is such that np is an integer.) Use
Stirling's approximation to show that the probability of getting exactly np
successes is
1
P(n, np, p) ≒ ───────
＿＿＿＿＿
√2πnp(1-p)
n! ＿＿
Note that lim ───── = √2π.
n→∞ n+1/2 -n
n e
--
2 2 1 ψxavier13540
給定一個二次元(|R )上的開集 G，設 f: G →|R ∈ C 。考慮一 autonomous system
╭dx/dt = f(x)，若 ∀t ≧ 0，有φ (x°) ∈ K ⊆ G，其中 K 在 G 上 compact，則
╰x(0) = x° t
ω(x°) 只能是一定點、一週期軌道或連接有限個 critical point 的連通路徑，不會像三
次元一樣可能出現混沌(chaos)。此即為 ODE 動力系統中的 Poincaré–Bendixson 定理。
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