1169/DCP1206 Probability, Fall 2016
30-Dec-2015
Homework 5 Solutions
Instructor: Prof. Wen-Guey Tzeng
Scribe: Amir Rezapour
1. Suppose that the distribution of the diastolic blood pressure is normal f
Probability and Statistics
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Midterm
10:10 am 12:00 noon, 4/16/08
Remember to write down your id number and your name.
Please provide detailed explanations/derivations in your answers. Correc
Probability and Statistics
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Final Exam
Solutions
2
2
1. (8 points) For jointly normal X and Y with respective variances X and Y , and
correlation coecient , the contour of the joint density is
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Probability and Statistics
Final Exam
1:30 p.m. 3:20 p.m., 6/22/09
Remember to write down your id number and your name.
Please provide detailed explanations/derivations in your answers. Cor
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Midterm
Solutions
1. (8+8=16 points)
(a) Let C denote the event of correct receiving and let A be the event that a 1 is transmitted.
Then,
P (C )
= P (C |A)P (A) +
Probability and Statistic
1
Homework #13
Due: A.m. 09:00 , Fri., June. 18, 2010, in class
1. (30%) For an increasing function : R [0, ), Show that
P (X )
E [(X )]
.
()
hint : Write P (X ) as an integ
Probability and Statistic
1
Solution #12
1. (30%) Let X1 N (1, 1) and X2 N (2, 2) be two normal random variables. Dene
X=
1
with probability 3
with probability 2 .
3
X1 ,
X2 ,
Find PDF and MGF of the
Probability and Statistic
1
Homework #12
Due: A.m. 10:10 , Fri., June. 9, 2010, in class
1. (30%) Let X1 N (1, 1) and X2 N (2, 2) be two normal random variables. Dene
X=
1
with probability 3
with prob
Probability and Statistic
1
Solution #11
1. (30%) Prove that if E [Y |X ] = E [Y ], then X and Y are uncorrelated.
Proof :
E [XY ] = E E [XY |X ]
= E X E [Y |X ]
= E X E [Y ]
= E [X ] E [Y ].
= X and
Probability and Statistic
1
Homework #11
Due: A.m. 09:00 , Fri., May 28, 2010, in class
1. (30%) Prove that if E [Y |X ] = E [Y ], then X and Y are uncorrelated.
2. (a) (40%) Show that the correlation
Probability and Statistic
1
Solution #10
1. Find the PDF of
(a) (20%) Z = X + Y , where X and Y are independent exponential random variable
with common parameter .
Sol :
P (Z z ) = P (X + Y z )
P (X +
Probability and Statistic
1
Homework #10
Due: A.m. 10:10 , Wed., May 19, 2010, in class
1. Find the PDF of
(a) (20%) Z = X + Y , where X and Y are independent exponential random variable
with common p
Probability and Statistic
1
Solution #9
1. Let X and Y be two continuous random variables with joint pdf
c y 0, |x| + y 1
0 otherwise
fXY (x, y ) =
(a) (15%)Find c.
Sol :
c dxdy = 1
y 0, |x|+y 1
= c =
Probability and Statistic
1
Homework #9
Due: A.m. 10:10 , Wed., May 12, 2010, in class
1. Let X and Y be two continuous random variables with joint pdf
fXY (x, y ) =
c y 0, |x| + y 1
0 otherwise
(a) (
Probability and Statistic
1
Solution #8
1. (30%) Given independent exponential random variables X1 , X2 , , Xn with parameter 1 , 2 , , n . Show that Y = mincfw_X1 , X2 , , Xn is also an exponential
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Probability and Statistics
Midterm
Solutions
1. (10 points) This problem intends to test whether you know the total probability theorem
pX (x) =
pY (y )pX |Y (x|y ).
y
And this can be shown b
3. (5+5=10 points) In a digital communication system, the receiver knows that the
transmitter will send either one of the messages: 00101 and 10011 with equal
probability. During the not-so-perfect tr
1191/DCP1206 Probability, Fall 2017
Due: 3-Nov-2017 (Friday)
Homework 3
Instructor: Prof. Wen-Guey Tzeng
Scribe: Amir Rezapour
1. The side measurement of a plastic die, manufactured by factory A, is a
1169/DCP1206 Probability, Fall 2016
11-Nov-2016
Homework 3 Solutions
Instructor: Prof. Wen-Guey Tzeng
Scribe: Amir Rezapour
1. The distribution function of a random variable X is given by
0 x < 2
1/2
1169/DCP1206 Probability, Fall 2016
21-Oct-2016
Homework 2 Solutions
Instructor: Prof. Wen-Guey Tzeng
Scribe: Amir Rezapour
1. Suppose that two fair dice have been tossed and the total of their top fa
1169/DCP1206 Probability, Fall 2016
7-Oct-2016
Homework 1 Solutions
Instructor: Prof. Wen-Guey Tzeng
Scribe: Amir Rezapour
1. A box contains three red and five blue balls. Define a sample space for th
Section 1.1
Applying Set Theory to Probability
1.1 Comment: Mutually Exclusive Sets
A collection of sets A1, . . . , An is mutually exclusive
if and only if
A
Ai Aj = ,
B
i 6= j.
(1.1)
The word disjoi
Probability and Statistics
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Quiz II
Solutions
1. (10+10+10=30 points)
(a) We need to check whether the joint PDF factors, i.e. whether fXY (x, y ) =
fX (x)fY (y ). In this problem, we can see
Probability and Statistics
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Quiz 1
9:00 am 9:50 am, 3/26/2010
Remember to write down your id number and your name.
Please provide detailed explanations/derivations in your answers. Correct a
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Probability and Statistics
Quiz 1
9:00 am 9:50 am, 3/26/2010
Remember to write down your id number and your name.
Please provide detailed explanations/derivations in your answers. Correct a
Probability and Statistics
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Quiz II
Solutions
1. (15 points) We must have that
f (x)dx
2. (8 + 7 = 15 points) (a) P (X < 3) =
1
2.
31
0 10 dx
= 1, which implies C = 3/8.
=
3
10 ,
and (b) P (3
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Probability and Statistics
Quiz II
9:00 am 9:50 am, 5/21/09
Remember to write down your id number and your name.
Please provide detailed explanations/derivations in your answers. Correct an
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Probability and Statistics
Quiz 1
Solutions
1. (10 points) Show that (S T ) T c = S T c .
Either by two-way reasoning or applying (S T ) T c = (S T c ) (T T c ) can easily prove
it. Use only
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Probability and Statistics
Quiz 1
9:00 am 9:50 am, 3/26/09
Remember to write down your id number and your name.
Please provide detailed explanations/derivations in your answers. Correct ans
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Probability and Statistics
Midterm
Solutions
1. (10 + 10 = 20 points) Assume X is a random variable with a uniform PMF on the integers
from -2 to 2. Dene W = X 2 . Please nd the following.
(a