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Solution for 36217
Wanjie Wang Teacher: Jiashun Jin September 30, 2009
If there is any question, please contact me at wwang@stat.cmu.edu 1. Denitions and expressions (a) the event that stands for event A happens but event B does not happen A B c or A/B (b
STAT 36-217, HW 2, due Thursday 09/10/2009, 10:30 AM 1. Suppose P (B ) > 0. Show that P (A B |B ) = P (A|B ). 2. Roll two fair dice, what is the probability that the sum is 2, 3, . . . , 12? 3. Two dice are rolled. What is the probability that at least on
HW 1, due Thursday 09/03/2009, 10:00 AM 1. A box contains 3 marbles: one red, one green, and on blue. Consider an experiment that consists of taking one marble from the box then replacing it in the box and drawing a second marble from the box. What is the
STAT 36-217, HW 3, due Thursday 09/17/2009, 10:30 AM 1. Suppose that we have n coins such that if the ith coin is ipped, heads will appear with probability i/n, i = 1, 2, . . . , n. When one of the coins is randomly selected and ipped, it shows heads. Fin
STAT 36-217, HW3, due Thursday 09/27/2011, 11:50 AM
1. Toss a fair coin independently for three times. Let Ai be the event that we get the
same outcome in the two tosses excluding the i-th one, 1 i 3 (i.e. A1 is the event
that we get the same outcomes in
36-217 Mini Exam 7 Solution
1
Draw the reion where the joint PDF is nonzero
Figure 1: Region of (X,Y) where the joint PDF is nonzero
2
Find c
1
x
fX,Y (x, y )dydx = 1
0
1
x
1
c
0
0
0
1
cy |x dx =
0
dydx =
0
c
c 21
x |0 =
=
2
2
c=
1
cx dx
0
1
2
3
Find E[Y]
STAT 36-217, HW 5, due Tuesday 10/18/2011, 11:50 AM
1. (30 pts). Given two random variables X and Y , each of them takes values from
cfw_1, 2, 3. The joint PMF of X and Y are given by p(1, 1) = 1/9, p(2, 1) = 1/3, p(3, 1) =
1/9, p(1, 2) = 1/9, p(2, 2) = 0
Assignment 9 Solutions
Problem 1
(a) We know the joint distribution of X and Y :
fX,Y (x, y ) =
1/ , x2 + y 2 1
0
, otherwise
If X and Y were independent, then the conditional distribution of X given Y would depend
only on X and not contain Y . The condit
STAT 36-217, HW 11, due Tuesday 12/06/2011, 11:50 AM
1. (10 pt). Cherno inequality. Fix s > 0. Show that for any random variable X such
that E [esX ] < ,
P (X a) esa E [esX ].
(Hint: apply Markov inequality to Y = esX ).
2. (10 pt). A sequence of random v
36-217 Homework 10 Solutions
Page 1 of 4
1. Suppose X Normal(0, 1). Then
1
2
fX (x) = ex /2
2
(a) Let Y = eX and h(y ) = ln y .
1
fX (h(y )| dh(y) | = fX (h(y ) y
dy
0
fY (y ) =
2
1
e(ln y ) /2
y 2
=
0
y>0
else
y>0
else
(b) Let Z = |X |. Two cases:
(1):
36-217 Mini Exam 9 Solution
1. X Unif(1, 1)
fX ( x) =
1
1(1)
0
=
1
2
if 1 < x < 1,
otherwise
Figure 1: PDF of X Unif(-1,1)
2. Y = log(X + 2)
(a) Range of Y
Since logarithm is a monotone increasing function,
log(1 + 2) < log(X + 2) < log(1 + 2)
0 < Y < log
36-217 Homework 4 Solutions
Page 1 of 7
1. (a) Recall the formula for marginal PMF:
pX (x) = P(X = x)
=
P(X = x, Y = y )
y
=
pX,Y (x, y )
y
From the given joint PMF, we know X takes values cfw_1, 2, 3, 4, 5.Therefore
pX (1) =
pX,Y (1, y )
y
= pX,Y (1, 1)
STAT 36-217 Probability Theory and Random Processes
Practice Final Exam
1. Mini exam 3, 6, 7.
2. For a regular deck of cards (without jokers), what is the change that in a hand of ve cards,
you have a full house?
3. A machine has two components A and B .
HW 1, due Tuesday 09/13/2011, 10:30 AM
Reading assignment: BT Section 1.1-1.2 and Lecture Note 1 (no need to hand in, but highly
recommend).
1. (20 points). Let S , T , U be three events. Find expressions for the events that
(a) at least one event occur.
36-217
Probability Theory and Random Processes
Mini Exam 2
1
Given a regular 52-card deck. Consider two experiments as follows.
In Experiment 1, we draw one card from the deck. Let A be the event that you get an Ace,
and let H be the event that you get a
STAT 36-217, HW 2 Solutions, due Tuesday 09/20/2011, 11:50 AM
Read Section 1.3-1.5 in BT.
1. (20). Assume that each child is equally likely to be born as a boy or a girl. Consider
a family that has three children.
What is the probability that the family
Assignment 3 Solutions
Problem 1
The events of interest are A1 , A2 , A3 where Ai , i = 1, 2, 3, is the event that we get the same
outcome in the two tosses excluding the ith one. Then A1 = cfw_HHH, T HH, HT T, T T T , A2 =
cfw_HHH, HT H, T HT, T T T , an
Mini - Exam 3 solutions
September 29, 2011
1 (20pt) Find the PMF of X in terms of p
It is similar to the standard negative binomial formulation (r,p) , but since two teams are playing
to determine the winner, theres a restraint on the range of X and we al
36-217 HW 11 Solution
Question 1.
Cherno inequality: Fix s > 0. Show that for any random variable X such that
E [esX ] < , P (X a) esa E [esX ].
Suppose Y = esX . Then, logY = sX X = logY /s.
By Markov inequality, we know that:
P (Y c) E (cY ) , for some
36-217 Mini-exam 1 solutions
1. (20pt) What is the probability that he gets the job?
P(Getting the job)
=
P(Getting the job AND Excellent recommendation)
+
P(Getting the job AND Strong recommendation)
+
P(Getting the job AND Moderately good recommendation
STAT 36-217, HW 2, due Tuesday 09/20/2011, 11:50 AM
Read Section 1.3-1.5 in BT.
1. (20). Assume that each child is equally likely to be born as a boy or a girl. Consider
a family that has three children.
What is the probability that the family has two bo
36-217 Homework 1 Solutions
1. (a)
(b)
(c)
(d)
Page 1 of 3
ST U
ST U
(S T U )c or by De Morgans law (S c T c U c )
(S T U )c or by De Morgans law (S c T c U c )
2. For simplicity, assume S , T , and U are nonempty sets. Recall that to show sets A and B ar
36-217 Mini Exam 5 Solution
1
Draw fX (x)
Figure 1: Plot of fX (x)
The probability density function is symmetric with respect to the y-axis (x = 0) and thus fX
is an even function.
2
Find the constant c
fX (x)dx = 1
0
1
c(1 x)dx = 1
c(1 + x)dx +
1
0
1
1
c
STAT 36-217, HW 6, due Tuesday 10/25/2011, 11:50 AM
1. (20 pts). Suppose X is the standard uniform random variable (i.e. uniform over the
interval (0, 1); we usually denote it by U (0, 1).
(a) Calculate var(X )
(b) Calculate E [X n ], n 1, and evaluate it
STAT 36-217, HW 7, due Thursday 10/22/2009, 10:30 AM
PLEAE USE THIS AS THE COVER PAGE
Your Name:
1
1. Let X and Y be independent random variables with means x and y and variances 2 2 x and y . Show that
22 2 2 Var(XY ) = x y + 2 x + 2 y . y x
2. Let X and
Solution for 36217
Wanjie Wang Teacher: Jiashun Jin October 22, 2009
If there is any question, please contact me at wwang@stat.cmu.edu 1. (10 Points) Let X and Y be independent random variables.
2 E [Y 2 ] = V ar(Y ) + (EY )2 = y + 2 y
Because X and Y are
STAT 36-217, HW 6, due Thursday 10/15/2009, 10:30 AM
PLEAE USE THIS AS THE COVER PAGE
Your Name:
1
1. What is the value of (n + 1/2)? What is the value of (n)? Use these to nd (6.5) and (11). 2. Suppose X is distributed as exponential with parameter = 2.