課程名稱︰微積分乙下
課程性質︰大一共同必修
課程教師︰陳其誠
開課系所︰醫學院、公衛學院等
考試時間︰2006.6.13
試題 :
Please note that this exam contains nine problem sets written in two pages
together with an one-page table. Write down your answers on the answer sheet.
You should include all the necessary calculations and reasoning. You can skip
the arithmetic and just give your answer in a form like
9 10
Σ ( )(0.4)^k(0.6)^(9-k).
k=5 k
1. Suppose plants with genotype BB and Bb are tall and plants with genotype bb
are short. We cross two plants with genotype Bb and produce N offspring. It
is known that an offspring is short with probability 1/4, tall with
probability 3/4, and we assume that heights of different offspring are
independent of each other. Let S be the random variable giving the number
of short offspring.
(a) Find the probability Pr(S = 5).
(b) Determine the mean μ(S) and the variance σ^2 (S). (5 points each)
2. Recall that if an event occurs at a probabilistic rate λ then the
probability that it occurs in the short time Δt is approximately λΔt.
More precisely, if p(Δt) is the probability that an event occurs in time
Δt, then
lim p(Δt)/Δt = λ
Δt→0
Suppose that some species of insect dies with probabilistic rate λ = 0.1
(per year), and let P(t) be the probability that an insect still alive at
time t.
(a) Show that the function P(t) satisfies the differential equation
,
P (t) = -0.1P(t) (5 points)
(b) What is the average life span of this species? (5 points)
3. A light bulb blows out with probability 0.005 each day. What is the
probability that it does not blow out for at least the first 365 days.
(8 points)
4. In each of the following, find and sketch the Poisson distribution
associated with the given probabilistic rate λ and duration t, and use it
to compute the requested probability.
(a) Phone calls arrives at a rate of λ = 0.1 each hour. What is the
probability that there are more than 6 calls in 8 hours.
(b) Molecules leaves a cell at rate λ = 0.2 each second. What is the
probability that three or fewer have left by the end of the third second
(5 points each)
5. In the following, use the attached table to calculate the requested
probability for the normal distribution N(μ,σ^2) with mean = μ and
variance = σ^2.
(a) Pr(Z ≦ 6) with μ = 11.25, σ^2 = (2.823)^2.
(b) Pr(8 ≦ Z ≦12) with μ = 10, σ^2 = 16. (5 points each)
_
6. In the following, let X = 1/n (X1+...+Xn) where X1,..., Xn are independent
and identically distributed random variables, and apply Central Limit
Theorem to find the suitable normal distribution N(μ,σ^2) to approximate
_
the distribution of X.
(a) Each Xi is a binomial distribution with p = 0.3 and n = 100.
(b) Each Xi is the outcome of throwing a fair dice and n = 100.
(6 points each)
7. Suppose we throw a dice 100 times and for exactly 40 times the outcome is
even (i.e. one of 2,4,6). Assume that the true probability foe getting an
even outcome is p.
(a) Find the likelihood function L(p).
(b) Determind the maximal likelihood.
(c) To determine the 95% confident interval we need to determine the
corresponding confident limits pl(l下標) and ph(h下標). Write down the
condintion that pl(l下標) satisfies. (5 points each)
8. A coin is flipped 400 times and comes out heads 100 times. It is thought
that the coin is fair. To test if the data provide significant evidence to
reject this hypothesis, we use the two tailed test.
(a) Write down the conditio, in terms of probability, for the test.
(8 points)
(b) Use some normal distribution as the approximation of your distribution,
and calculate the significance level. (7 points)
9. A medical study is conducted to estimate the proportion of people suffering
from season affected disorder. We need to determine how many people should
be surveyed to be at least 99% sure that the estimate is within 0.02 of the
true value.
(a) Suppose we survey N people and let X1,..., XN be the outcome of the
survey such that Xi = 1 (resp. 0) if the i'th surveyed person is
_
affected (resp. not affected). Let X = 1/N (X1+...+XN) and let μ be the
true proportion of affected people. Show that our condition is equiva-
lent to
_
Pr(-0.02/σ≦ (X - μ)/σ ≦ 0.02/σ) ≧ 0.99,
_
where σ^2 is the variance of X.
(b) Write σ^2 in terms of μ and N.
(c) Use a suitable normal distribution to get a condition in terms of σ
and N.
(d) Use the fact that the maximal of the fuction f(x) is 1/4 to determine
the least value of N. (10 points total)
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