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 disjoint is sometimes used as a synonym
for mutually exclusiv
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 integral, and then use increasing (x) () for x )
and nonnega
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 random variable X .
Sol : By total probability theorem,
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 probability 2 .
3
X1 ,
X2 ,
Find PDF and MGF of the random
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 Y are uncorrelated.
2. (a) (40%) Show that the correlat
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 coecient | 1.
hint : Problem1, HW4
(b) (30%) In (a), w
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 + Y z |Y = y )fY (y ) dy
=
(total probability theorem)
y
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 parameter .
(b) (20%) Z = eX , where X is a standard Gau
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 = 1.
(b) (15%)Are X and Y independent? Justify your answ
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) (15%)Find c.
(b) (15%)Are X and Y independent? Justify y
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
random variable with parameter = n i .
i=1
Proof : We r
Probability and Statistic
1
Homework #8
Due: A.m. 10:10 , Wed., May 5, 2010, in class
1. (30%) Given i.i.d. exponential random variables X1 , X2 , , Xn with parameter
1 , 2 , , n . Show that Y = mincfw_X1 , X2 , , Xn is also an exponential random
variabl
Probability and Statistic
1
Solution #7
1. (50%) For a nonnegative continuous random variable X, show that
P (X > x) dx.
E [X ] =
0
Proof :
x=0
t=x
fX (t) dt dx
P (X > x) dx =
0
t
t=0
x=0
=
fX (t) dx dt (change integration order)
=
tfX (t) dt
t=0
= E [X ]
Probability and Statistic
1
Homework #7
Due: A.m. 10:10 , Wed., Apr. 28, 2010, in class
1. (50%) For a nonnegative continuous random variable X, show that
P (X > x) dx.
E [X ] =
0
hint : Compare with problem 2 in HW4.
2. (50%) Let X be an exponential rand
Probability and Statistics
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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) + P (C |Ac )P (Ac )
= (1 )p + (1 )(1 p)
= 1 .
(b) Let B b
nctuee09
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. Correct answers without any
explanations will not be given
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 X and Y are indeed independent by
direct inspection tha
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 answers without any explanations
will not be given any c
nctuee10
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 answers without any explanations
will not be given any c
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 < X < 8) =
81
3 10 dx
=
3. (20 points)
fY (y ) = fY |=
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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 answers without any
explanations will not be given any cr
nctuee09
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 Venn diagram gets partial credits.
2. (15+10+10=35 poin
nctuee09
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 answers without any explanations
will not be given any cre
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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) pW |X (w|x).
p(4| 2) = 1, p(1| 1) = 1, p(0|0) = 1, p(
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Probability and Statistics
Midterm
1:30 p.m. 3:20 p.m., 4/20/09
Remember to write down your id number and your name.
Please provide detailed explanations/derivations in your answers. Correct answers without any explanations
will not be given an
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 transmission, each binary bit independently has
the proba
nctuee08
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 by
pX (x)
= P (X = x)
=
P
cfw_X = x Y = y
y
=
P (X = x,
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. Correct answers without any explanations
will not be given an
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 a circle if and only if
X = Y and = 0.
This can be exp
Probability and Statistic
1
Solution #6
1. Given a joint PMF
PXY (x, y ) = C 2|xy|
x N, y Z \ cfw_ 0
(a) (10%) Find C.
Sol :
C 2|xy| = 1
xN y Z\cfw_0
2xy = 1
= 2C
x=1 y =1
= 2C
x=1
2x
1
=1
1
1
= C =
2
x=1 2x 1
.
(b) (20%) Find the conditional probability