課程名稱︰機率導論
課程性質︰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）︰2013/04/23
考試時限（分鐘）：13:20PM-15:10
試題 :
Intrductory Probability 機率導論
期中考
考試時間: 13:20PM-15:10 Tuesday, April 23rd, 2013
題組一 : 定義、名詞、基本演算 (200points)
1. (45 points) Let
Q = {(x,y) ∈ R^2 : x and y are integers, x≧0, y≧0 and x+y≦2}.
(I suggest you to draw a picture of Q first.) There are random variables X and
Y such that
╭ x + y
│─── if (x,y) ∈ Q,
p_X,Y(x,y) = ﹤ 8
│ 0 otherwise.
╰
Determine (a) (15 points) E(X + Y), (b) (15 points) Var(X + Y) and
(c) (15 points) Whether X and Y are independent.
2. (30 points) Suppose A,B,C,D are independent events.
Compute (a) (15 points) P((A∪B)∩(C∪D)) and (b) (15 points) P((A*B)∪(C∪D))
in terms of P(A), P(B), P(C), and P(D). (Note that A*B = (A\B)∪(B\A) and
c
A\B = A∩B .)
3. (30 points) Let k and ι be two independent random digits between 0 and 9.
Denote by X = k + ι and Y = k˙ι. Compute (a) (10 points) E[Y] and
(b) (20 points) Cov(X,Y). (Note that Cov(X,Y) = E[XY] - E[X]E[Y].)
4. (45 points) We draw a random vector (X,Y) non-uniformly from the
right-angled triangle with vertices (0,0), (1,0), (1,1) in the following way:
First we draw X uniformly on [0,1]. Then, given X = x we draw Y uniformly
on [0,x].
(a) (10 points) Give the conditional pdf of Y given X = x. Specify where this
conditional pdf is 0.
(b) (15 points) Find the joint pdf of X and Y.
(c) (20 points) Calculate the pdf of Y and sketch its graph.
5. (50 points) The number of accidents that a person has in a given year is a
Poisson random variable with mean λ. However, suppose that the value of λ
varies in from person to person. Assume the proportion of the population having
a value of λ less than x is equal to 1 - exp(-x). If a person is chosen at
random, what is the probability that he will have
(a) (25 points) 0 accidents in a year.
(b) (25 points) 0 accidents in a year given he had no accident the preceding
year.
Hint: Let L denote the number accidents of this person and determine
P(L = 0|λ = x).
題組二:應用 (75 points)
6. (30 points) A sieve(濾網) with diameter d is used to separate a large number
of blueberries into two classes: small and large. Suppose the diameters of the
blueberries are normally distributed with an expectation μ = 1 (cm) and a
standard deviation ρ = 0.1 (cm).
(a) (15 points) How large should the diameter of the sieve be, so that the
proportion of large bluberries is 30%?
(b) (15 points) Suppose that the diameter is chosen such as in (a). What is the
probability that out of 1000 blueberries, fewer than 280 end up in the large
class?
7. (45 points) A lawyer for the prosecution in a murder trial observes that a
suspect's fingerprints match those found at the crime scene and that there is
a 1 in 50,000 chance of such a match occurring by chance. They argue that this
means there is a 1 in 50,000 chance of the suspect being innocent.
(a) (15 points) What do you think of this argument?
(b) (15 points) Suppose that the city has a population of 1 million and the
murderer is assumed to be one of these. If there is no evidence against the
suspect apart from the fingerprint match, is reasonable to regard him as a
randomly chosen citizen. Under this assumption what is the probability that he
is innocent?
(c) (15 points) Indicate briefly what element of the calculation would change
if you were told that the victim almost certainly knew his killer and that the
suspect's name had been found in the victim's adress book.
Hint: 回答此題時，建議使用 I 代表事件{the suspect is innocent} and F 代表事件
{the fingerprints of the suspect match those at the crime scene}及適當的數學式
來回答此問題。
題組三 : (120 points)
8. (40 points) Let x be a Poisson random variable with parameter λ and Y a
geometric random variable with parameter p (i.e., P(Y = y) = [(1-p)^y-1]p).
Assume that X and Y are independent. Compute P(Y > X).
9. (40 points) Suppose r(x) is differentiable and {x:r'(x) = 0} is finite. Show
that the density function of Y = r(X) is given by
f(x)
Σ ──── .
{x:r(x)=y} |r'(x)|
10. (40 points) Let X_n have a geometric distribution with parameter p = λ/n.
(Refer to Q8.) Determine a such that
X_n
P(─── > x) does not go to 0 or 1 as n→∞.
n^a