推 tdk4:永和市長? 07/10 19:21
課程名稱︰機率
課程性質︰資訊工程學系 系定必修
課程教師︰洪一平
開課系所︰資訊工程學系
考試時間︰2006年6月22日
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 equal the number of alpha particle emissions of carbon-14 that
are counted by a Geiger per second. Assume that the distribution of X is
Poisson with mean 16. Ler W equal the time in seconds before the seventh
count is made.
(a) Give the distribution of W.
(b) Find P(W≦0.5).
2.(10%) An automobile repair shop makes an initial estimate X(in thousnds)
needed to fix a car after an accident. Say X has the p.d.f.
f(x)=2e^[-2(x-0.2)] , 0.2< x < 無窮大
Give X=x, the final payment Y has a uniform distribution between x-0.1 and
x+0.1. What is the expected value of Y?
3.(10%) Find the distrubution of W =X^2 when
(a) X is N(0,4),
(b) X is N(0,δ^2).
4.(10%) Let X be N(0,1). Find the p.d.f of Y=|X|, a distribution that is often
called the half normal.
5.(10%) Let X equal the maximal oxygen intake of a human on a treadmill, where
the measurements are in milliliters of oxygen per minute per kilogram of
weight, Assume that of a particular population the mean of X is μ=54.030 and
_
the standard deviation is δ=5.8. Let X be the sample mean of a random
_
sample of size n=47. Find P(52.761≦ X ≦54.453), approximately.
6.(10%) Out of 50,000,000 instant winner lottery tickets, the proportion of
winning tickets is p. Each day, for 20 consecutive days, abetter purchased
tickets, one at a time, until a winning ticket was purchased. The numbers of
tickets that were purchased each day to obtain the winning ticket were
1 26 19 6 6 1 2 3 1 23
19 3 6 8 4 1 18 34 1 8
By making reasonable, find the maximum likelihood estimate of p based on
these data.
7.A leakage test was conducted to determine the effectiveness of a seal
designed to keep the inside of a plug air tight. An air needle was inserted
in the plug and this was placed under water. The pressure was then increased
until leakage was observed. Let X equal the pressure in pounds per square
inch. Assume that the distribution of X is N(μ,δ^2). Using the following
n=10 observations of X,
3.1 3.3 4.5 2.8 3.5 3.5 3.7 4.2 3.9 3.3
Note:
3.1^2 +3.3^2 +4.5^2 +2.8^2 +3.5^2 +3.5^2 +3.7^2 +4.2^2 +3.9^2 +3.3^2 =130.52
(a) Find a point estimate of μ.
(b) Find a point estimate of δ.
(c) Find a 95% one-sided confidence interval for μ that provides an upper
bound for μ.
8.Let X be a random variable with the following probability distribution:
f(x)= (θ+1)x^θ , 0≦ x ≦1
0 ,otherwise
(a) Find the maximum likelihood estimator of θ, based on a random sample of
size n.
(b) Find the moment estimator of θ.
9.The joint probability distribution is
x -1 0 0 1
y 0 -1 1 0
f(X,Y) 1/4 1/4 1/4 1/4
(a) What is the correlation between x and y?
(b) Are they independent? Why?
10.Let X and Y represent concentration and viscosity of a chemical product.
Suppose X and Y have a bivariate normal distribution with δx = 4, δy = 1,
μx = 2, μy = 1. Draw a rough contour plot of the joint probability density
function for each of the following values of ρ
(a)ρ = 0
(b)ρ = 0.8
(c)ρ = -0.8
註.題目卷有附相關table,以供查閱,此略.
※ 編輯: greensunlife 來自: 140.112.250.104 (07/08 13:28)