NAME:
Math 3DC3 Test # 2
March 20, 2015
ID #
Instructions: This exam consists of 3 questions in 6 pages. Indicate your answers clearly
in the appropriate places. Justify your answers in order to receive full credit. The textbook
and the class notes are al
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Test # 2
FIRST (given) NAME:
Math 3DC3
March 20, 2009
ID # :
Instructions: This exam consists of 3 questions in 7 pages. Indicate your answers clearly in the
appropriate places. Justify your answers in order to receive full credit. Use
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MATHEMATICS 3DC3
McMaster University Final Examination
Day Class
Duration of Examination: 3 hours
Dr. S. Alama
April 2009
THIS EXAMINATION PAPER INCLUDES 15 PAGES AND 4 QUESTIONS. YOU
ARE RESPONSIBLE FOR ENSURING THA
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Test # 1
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Math 3DC3
October 7, 2010
ID # :
Instructions: This exam consists of 3 questions in 6 pages. Indicate your answers clearly
in the appropriate places. Justify your answers in order to receive full credit. Us
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Test # 2
FIRST (given) NAME:
Math 3DC3
November 4, 2010
ID # :
Instructions: This exam consists of 3 questions in 6 pages. Indicate your answers clearly
in the appropriate places. Justify your answers in order to receive full credit. U
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MATHEMATICS 3DC3
McMaster University Final Examination
Day Class
Duration of Examination: 3 hours
Dr. D. Pelinovsky
December 8, 2010
THIS EXAMINATION PAPER INCLUDES 16 PAGES AND 5 QUESTIONS. YOU
ARE RESPONSIBLE FOR E
NAME: - Math 3DC3 Test # 1
ID # : January 29, 2015
Instructions: This exam consists of 3 questions in 6 pages. Indicate your answers clearly
- in the appropriate places. Justify your answers in order to receive full credit. Use the backs
of pages if neces
Test # 2 / Math 3DC3 -2-
1. Consider the tripling map F(3:) = 33(mod1) on the interval I = [0, 1]. I
(A) [2 pts.] Use ternary expansions
00
E[0,1]: x=Z%—, where sn€{0,1,2}, nEN,
n=1
and compute the ternary expansion of F(a‘).
(B) [4 pts.] Find all periodi
NAME:
Math 3DC3 Test # 1
January 29, 2015
ID #
Instructions: This exam consists of 3 questions in 6 pages. Indicate your answers clearly
in the appropriate places. Justify your answers in order to receive full credit. The textbook
and the class notes are
LAST (Family) NAME:
Test # 1
FIRST (given) NAME:
Math 3DC3
February 6, 2009
ID # :
Instructions: This exam consists of 2 questions in 7 pages. Indicate your answers clearly
in the appropriate places. Justify your answers in order to receive full credit. U
Mathematics 3DC3
ASSIGNMENT 3 Solutions
3. Consider the map: Cc (x) = c cos(x).
Use XPPAUT to draw:
(i) an orbit diagram for c [6, 0], and x [6.2, 6.2], starting each orbit at the critical point x = 0.
Explain the dramatic bifurcation that occurs near c
Mathematics 3DC3
ASSIGNMENT 4 Solutions
4(a) Consider the map: f (x) = x2 + 1.
2
y
4
Newtons method for f(x)=x +1
3
2
1
0
-1
-2
-3
-4
x0=0.3
-4
-3
-2
-1
0
x
1
2
3
4
Figure 1: Since there are no real roots of f (x) = 0, there are no xed points of the Newto
Empirical Rule for X
Consider a sample of size 'n' from a population with mean
Suppose X is normal (or approximately normal), with
and standard deviation
x
, and
/ n
x
(This would be the case if the population is normal or if the sample size is large)
Fin
(c) Notice that " X within 2 of the population mean", means that X falls in the following
interval
128
P(
130
X
132
)
X
P
X
P(
P( Z
)
Z
)
P( Z
)
135
Example 2: The scores on a Standardized Reading Test are known to be normally distributed
with a mean of =
We now calculate 1000 values of X calculated from 1000 samples each of size n = 36.
c50: 2 4 5 6
c51: .1 .2 .4 .3
MTB > rand 1000 c1-c36;
SUBC> discr c50 c51.
MTB > rmean c1-c36 c37
The 1000 x 's are in c37
MTB > hist c37;
SUBC> incr .2.
Histogram of C37
Example: Suppose that a certain temperature reading X in oF are normally distributed with,
65, X = 10.
Let Y = the temperature reading in oC.
Since oC = (5/9)oF - (160/9),
Y = (5/9)X - (160/9)
Also, since X is normal, Y is normal with
mean
variance
Y
= (5
4. Union: A or B [A
S
B]
The union of two events A and B is the
event that A or B or both occur. It is made
up of all the outcomes which are in A
together with all the outcomes which are
in B.
Addition Rule:
P(A or B) = P(A) + P(B) - P(A and B)
To illustr
Example 3 (continued)
S = cfw_(r,1), (r,2), (r,3), (b,1), (b,2), (b,3)
A = cfw_(r,1), (r,3), (b,1), (b,3)
["an odd number is drawn"]
B = cfw_(b,1), (b,2), (b,3)
["a black ball is drawn"]
(iii) A and B =
N ( A and B)
N (S )
P( A and B)
(iv) A or B =
P ( A
There are a number of equivalent ways of defining independence.
Two events A and B are said to be INDEPENDENT if any of the following equivalent
statements hold:
(a) P(A | B) = P(A)
(b) P(B | A) = P(B)
(c) P(A and B) = P(A)P(B)
Statement (c) is the MULTIP
We can now formalize the methods that we used above.
Conditional Probability:
The conditional probability of an event A given that another event B has occurred is defined
by
P( A and B)
P( A | B)
, where P( B ) 0
P( B)
Note: This definition implies that f