課程名稱︰微積分乙
課程性質︰共同必修
課程教師︰陳其誠
開課系所︰
考試時間︰2006/5/2
試題 :
Write down your answers on the answer sheet. You should include all the
necessary calculations and reasoning. This exam contains 7 problem sets written
in two pages.
(1) Calculate the mean (expectation) μ(X) (=E(X)) and the variance σ2(X)
(=Var(X)) of the random variable X in the following cases (10points each):
(a) suppose the random variable X has values {1,2,3,4} with Pr(X=k) =k/10
for k=1,2,3,4.
(b) Suppose the density function f satisfies
4χ3, if χε[0,1];
f(χ) ={
0, otherwise.
(c) Suppose the density funxtion f satisfies
e(-x),if χ≧;
f(χ) ={
0,otherwise.
(d) Suppose X = Y+Z+W, where Y,Z,W are independent random variables with
μ(Y)=1/3,μ(Z) =2/3, μ(W)=1, σ2(Y)=1/9,σ2(Z)=4/9,σ2(W)=1.
(2) A species of bird comes in three colors: red, blue amd green. 20%are red,
30% are blue, and 50% are green. Females prefer red to blue and blue to
green, and they mate with the best they find. In fact females pick the
better of the first two male they meet. What is the probability that a
female mates with a green bird? What did you have to assume about the
independence (10 points)?
(3) Suppose that I undergo a medical test for a relatively rare cancer. My
doctor tells me that, the cancer has a incidence of 1% among the general
population. Extensive trials have shown that the reliability of the test
is 75%. More precisely, although the test does not fail to detect the
cancer when it is present, it gives a positive result in 25% of the cases
where no cancer is present. When I am tested, the test produces a positive
diagnosis. Given the result of the test, what is the probability that I
have the test? (12 points)
(4) We toss a coin repeatedly until the first head show up. Assume that the
probability of heads is 1/3. Let Y be the random variable that counts the
number of trials until the first head show up. What id the probability
Pr(Y=k)? (10 points)
(5) Suppose that X1,X2,X3 are independent random variables where for k=1,2,3
1 ,if χ≧1;
Pr(Xk≧χ)={χ,if χε[0,1];
0 ,if χ< 0.
Define Y= max(X1,X2,X3). Determine P(Y≦y) and find the density function
y
of Y. Namely, find f(y) such that P(Y≦y) = ∫ f(t)dt (10 points).
0
(6) In tropical regions, caterpillars suffer extremely high parasitism. Suppose
a caterpillar is attacked by a parasitoid with probability 15% per day. But
the caterpillars also have some chance of eliminating their attacker, thus
becoming a caterpillar again, with probability 3%per day. Find the
probabiility that a caterpillar after a long period of time (10 points).
(7) We have learned from the text book that if two random variables X and Y are
independent then their covariance is zero. Conversely, if two random
variables X and Y have zero covariance, then are there necessary idependent
? If you say yes, give a reason; if you say no, give an example to
illustrate it (8 points).
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