※ [本文轉錄自 NTU-Exam 看板]
作者: prolegend (黑誠) 看板: NTU-Exam
標題: [試題] 94下 張鎮華 微積分乙 3rd考試
時間: Fri Jun 30 11:17:51 2006
課程名稱︰微積分乙
課程性質︰大一共同必修
課程教師︰張鎮華
開課系所︰醫理公衛生科院等
考試時間︰2006.06.27(二) 10:20-12:10
是否需發放獎勵金：是，謝謝！
（如未明確表示，則不予發放）
試題 :
Examination 3 (Calculus B, second semester)
2006.6.27 (Gerard J. Chang)
1.(15%) A stardard deck of cards consists of 52 cards, arranged in four suits,
each with 13 difficult values. In the game of poker, a hand of five cards
drawn at random from the deck without replacement.
(a) How many hands are possible?
(b) How many hands consist of exactly one pair?
(c) How many hands consist of exactly two pairs?
(d) How many hands are flush (five cards of the same suit)?
(e) How many hands are four of a kind (four cards of the same value plus one
other card)?
2.(10%) An urn contains four green, five blue and six red balls. You take
three balls out of the urn without replacement.
(a) Determine the sample space Ω and find |Ω|.
(b) What is the probability that all three balls are of different colors?
(c) What is the probability that at least one of the three balls is green?
3.(10%) A test for the HIV virus shows a positive result in 99% of all cases
when the virus is actually present and in 5% of all cases when the virus is
not present (a flase positive result). If such a tst is administered to a
randomly chosen individual, what is the probability that the test result is
positive? What is the probability that a person is infected given that the
result is positive? Assume that the prevalence of the virus in the
population is 1%.
4.(15%) Consider a sequence of independent Bernoulli trails, namely, a random
experiment of repeated trails where each trail has two possible outcomes,
success or failure, and the trails are independent. Denote the probability
of success by p. Let X counts the number of trails until the first success.
(a) Determine P(X = k) for k = 1,2,3,...
(b) Compute the probability of no success in the first k trails, i.e.,
P(X > k).
(c) Compute the probability of no success in k trails following n
unsuccessful trails, i.e., P(X > n+k | X > n).
(d) Compute EX.
(e) Compute var(X).
5.(15%) Suppose that X_1, X_2x...,X_n are independent and unifromly distribute
over (0,1). Define
X = man{X_1. X_2,...,X_n }
(a) Determine the distribution of X, i.e, F(x) = P(X ≦ x)for x屬於R.
(b) Fine the density function f for X.
(c) Compute EX.
(d) Compute var(X).
(e) Compute lim P(X > x/n).
n→∞
6.(10%) Suppose X_1, X_2,... X_n are i.i.d. with
┌ 1, with probability p,
X_i│
└ 0, with probability 1-p.
_ 1 n
Set X_n = ─ Σ X_i.
n i=1
(a) Compute EX_i and var(X_i) for each i.
_ _
(b) Compute EX_n and var(X_n).
_
(c) Use Chebyshev's inequality to fine n so that X_n will differ from p by
less than 0.01 with probability at least 0.99.
7.(10%) Suppose we have un-fair coin, showing head with probability only 0.2.
Solve the following problems giving the solution in terms of the
distribution function F(x) of the standard normal distribution. For
instance, writing the answer as F(10) if fine.
(a) Toss the coin 625 times, use the central limit theorem to fine an
approximation for the probability of at least 300 heads.
(b) Find an approximation for the probability of exactly 125 heads.
8.(15%) Suppose we denote the sample by (X_1, X_2,...,X_n), where X_k is the
kth observation. Recall that
_ 1 n
the sample mean is X_n = ─ Σ X_k;
n k=1
2 1 n _ 2
the sample variance is S_n = ── Σ (X_k - X_n) .
n-1 k=1
(a) Compute the sample mean and the sample variance of the date: 6, 5, 9, 7,
8, 7, 8, 5, 7, 8, 7.
Suppose (X_1, X_2,..., X_n) are i.i.d. with each EX_k = μ and each
var(X_k) = σ^2.
_
(b) Compute EX_n.
_
(c) Compute var(Xn).
2
(d) Compute E(S_n ).