.
Activity
Two Congruent Halves
<f the figures that follow, draw a line from dot to dot to cut the figure in half in such a way that the two
.-es are congruent. The dotted line illustrates how the first one can be done.
-f
Can you think of a way that coul
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Fall 2010)
Problem Set 8
Due November 15, 2010
1. Oscar goes for a run each morning. When he leaves his house for hi
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Fall 2010)
Problem Set 11 Solutions
1. Check book solutions.
2. (a) To nd the MAP estimate, we need to nd the value
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Fall 2010)
Problem Set 1: Solutions
Due: September 15, 2010
1. (a) A B C
(b) (A B c C c ) (Ac B C c ) (Ac B c C) (Ac
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Fall 2010)
Problem Set 3 Solutions
Due September 29, 2010
1. The hats of n persons are thrown into a box. The person
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Fall 2010)
Problem Set 4: Solutions
1. (a) From the joint PMF, there are six (x, y) coordinate pairs with nonzero pr
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Fall 2010)
Problem Set 9 Solutions
1. (a) Yes, to 0. Applying the weak law of large numbers, we have
P(|Ui | > ) 0 a
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Fall 2010)
Problem Set 5: Solutions
1. (a) Because of the required normalization property of any joint PDF,
2
2
2
1=
6.001, Spring Semester, 20057Qu1'z I 7 Sample Solutions 2
Part 1: (20 points)
For each of the following expressions or sequences of expressions, state the value returned as the
result of evaluating the nal expression in each set, or indicate that the eval
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Fall 2010)
Problem Set 8: Solutions
1. (a) We consider a Markov chain with states 0, 1, 2, 3, 4, 5, where state i in
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Fall 2010)
Problem Set 6: Solutions
1. Let us draw the region where fX,Y (x, y) is nonzero:
y
2
1
y-x=z
0
1
2
x=2
x
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Fall 2010)
Problem Set 11
Never Due
Covered on Final Exam
1. Problem 7, page 509 in textbook
Derive the ML estimator
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Fall 2010)
Problem Set 5
Due October 18, 2010
1. Random variables X and Y are distributed according to the joint PDF
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Fall 2010)
Problem Set 7: Solutions
1. (a) The event of the ith success occuring before the jth failure is equivalen
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Fall 2010)
Problem Set 6
Due October 27, 2010
1. Random variables X and Y are distributed according to the joint PDF
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Fall 2010)
Problem Set 2
Due September 22, 2010
1. Most mornings, Victor checks the weather report before deciding w
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Fall 2010)
Problem Set 2: Solutions
Due September 22, 2010
1. (a) The tree representation during the winter can be d
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Fall 2010)
Problem Set 9
Due November 22, 2010
1. Random variable X is uniformly distributed between 1.0 and 1.0. Le
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Fall 2010)
Problem Set 10 Solutions
1. A nancial parable.
(a) The bank becomes insolvent if the assets gain R 5 (i.e
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Fall 2010)
Problem Set 3
Due September 29, 2010
1. The hats of n persons are thrown into a box. The persons then pic
Massachusetts nstitute of Technology
Department of Electrical Engineering Computer Science
6 041/6 431: Probabilistic Systems Analysis
(Fall 2010)
Problem Set 4
Due October 6 2010
1. Random variables X and Y have the joint PMF
pX Y (x, y) =
c(x + y ), if
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Fall 2010)
Problem Set 10
Due December 2, 2010 (in recitation)
1. A nancial parable. An investment bank is managing
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Fall 2010)
Problem Set 7
Due November 8, 2010
1. Consider a sequence of mutually independent, identically distribute
l
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100
Exam
1
Spring
2 013
- 5130-6:45pm
TR
Date:215113
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e font
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t
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Trapezoid
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Parallelogram
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-Opposite sides are congruent -One pair of opposite angles is congruent
-Opposite angles are congru
Lesson Title: Sum of the Angles of Any Polygon
Level: 5th Grade
Acceleration/Enrichment
Submitted by:
Materials needed:
Blank paper for each group member
A pencil to write with
Ruler or straight edge
Calculator
Protractor
Activity 1.4 Handout (Problems 1
Kite: A quadrilateral with two separate pairs of adjacent congruent sides
Possible Definition A:
A kite is a quadrilateral that has at least one pair of congruent opposite angles
Possible Definition B:
A kite is a quadrilateral that has perpendicular diag
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Completeeachproblem as necessory. how oll work or no credit wilt begiven.
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q uestions
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t he
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Spring 013
2
Name:
Showall work or no creditwill be given:
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f
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2. The measr:re gsn angle is 10o more than three times the measureof