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(MATH241)[2010](s)midterm~1365^_375.pdf
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HKUST
MATH 241 Probability
Midterm Examination Name:
27th March 2010 Student I.D.:
10:00C12:00 Tutorial Section:

Directions:
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Write your name, ID number, and tutorial section in the space provided above.

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DO NOT open the exam booklet until instructed to do so.

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Turn o. all mobile phones and pagers during the examination.

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This is a closed book examination.

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Unless otherwise speci.ed, numerical answers should be either exact or correct to 4 decimal places.

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You are advised to try the problems you feel more comfortable with .rst.

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You may write on both sides of the examination papers.

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You must show the working steps of your answers in order to receive full points.

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You must not possess any written or printed papers that contains information related to this examination.

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Cheating is a serious o.ense. Students caught cheating are subject to a zero score as well as additional penalties.


Question No. Points Out of
Q. 1 10
Q. 2 10
Q. 3 10
Q. 4 15
Q. 5 20
Q. 6 10
Q. 7 10
Total Points 85

Answer ALL questions on the space provided.

1. [ 10 Marks ] Please consider the following questions:
(a)
[5] Consider a urn of m indistinguishable red balls and n . m indistinguishable blue balls, where m<n . m. How many linear orderings are there in which no two red balls are next to each other?

(b)
[5] Consider a class of n students of which m are boys and n . m are girls, where m<n . m. How many linear orderings are there in which no two boys are next to each other?


2. [ 10 Marks ] Consider a class of n students, where n N = 365. Find the probability that
(a)
[5] exactly m students were born on Oct 1st, for m n;

(b)
[2] at least one student was born on Oct 1st.

(c)
[3] Hence, show that


n
. Cn (N . 1)n.m = Nn .
m
m=0
3. [10Marks]

(a) [5] Consider a bag of N white balls and M black balls. Now we draw a ball with replacement N times. Then prove that the probability that there are an odd number of black balls chosen in these N draws is
1.1 . (1 . 2p)N .,2
M
where p =.
N + M
(b) [5] If {En : n =1, 2,..., } is a decreasing sequence, i.e. E1 . E2 . , then prove that

lim P (En)= P ( . En).
n
n=1
4. [15Marks]If X Bin(n, p), where 0 <p< 1, then
(a)
[7] Show that as k goes from 0 to n, PX (X = k) .rst increases monotonically and then decreases monotonically, reaching its largest value when k is the largest integer less than or equal to (n + 1)p. (Hint: consider PX (X = k)/PX (X = k . 1)).

(b)
[8] Sketch the graph of the pmf of X against k when (i) n = 10,p =0.5, (ii) n = 11,p =0.5,


(iii) n =9,p =0.3, and (iv) n =9,p =0.8,
5. [ 20 Marks ] Consider a bag of N white balls and M black balls.
(a)
Now draw a ball with replacement, and let X be the random variable of

i. the number of black balls chosen in n draws;
ii. the numb