課程名稱︰微積分乙
課程性質︰系定必修
課程教師︰康明昌
開課學院：醫學院
開課系所︰醫學系
考試日期（年月日）︰97年6月17日
考試時限（分鐘）：120分鐘
是否需發放獎勵金：是，謝謝
（如未明確表示，則不予發放）
試題 :
每題15分
1.Draw a number among the three numbers 1,2,3.
Repeat this experiment three times. Let Xi be the number of the i-th draw
for i=1,2,3. Suppose that
P(Xi = 1) = 1/2 P(Xi = 2) = P(Xi = 3)= 1/4
Define a new random variable X by X=(X1+X2+X3)/3
(1)Find the probability mass function of X
(2)Find EX
(Hint: If the outcome is (1,2,1), then X=4/3.
You should find all the outcomes.)
2.Throw a coin until a HEAD turns up for the third time. Let p be the
probability that a throw results in a HEAD. Let X be the random variable of
the number of times we should throw in order to finish the game.
Write P(X = n)= A* (p^a)*[(1-p)^b]
when n >= 3 and A,a,b are real numbers. Find A,a,b.
3.The number of phone calls arriving at a switch board per hour is a Poisson
distribution with mean 3 calls per hour. Find the probability that at most
two calls arrive between noon and 3 p.m.
4.Let X be a continuous random variable with probability density function
f(x) = 0 , if x<0 or x>1
4x-4x^3 , if 0 <= x <= 1
Find var (X^2 + 3X)
5.Let Ω be the sample space Ω = {(x,y): 2 <= x <= 3, 1 <= y <= 2}
(2,2) (3,2)
┌─────┐
│ A0 │
│ ∕﹨ │
│ ∕ ﹨ │
│ ∕ ﹨ │
│∕ ﹨│
└─────┘
A1(2,1) A2(3,1)
(說明：A0 是在此正方形內部一點) A0(x,y)
The probability of a set B<Ω is the area of B
We will choose an arbitrary point A0(x,y) in Ω.
Define a random variable X to be the area of △A0A1A2
(1) Show that {X<= 1/6 } = { (x,y) ε(屬於)εΩ:1 <= y <= 4/3 }
(2) Find EX
(Hint : Find the cumulative distribution function and the probability
density function of X.)
6. Toss a fair coin 500 times. Let S500 be the random variable of the number
that the HEAD appears. Use the central limit theorem to approximate S500. If
P(249.5 <= S500<= 250.5 ~[10^ -3 ]*A
where A is a possitive integer, find A
(Hint : Use the table of the standard normal distribution.)
(ps.考試時有附一張標準的常態分配值的表)
7. (1) Give the definition that the random variables X1,X2,....,Xn,...
converges to α in probability when α ε(屬於) R(常數)
(2) State the Law of Large Numbers.
(3) State the Central Limit Theorem.
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