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Quiz 3 Oral Quiz
Oral Quiz
• There are 25 questions in total
are 25 questions in total
• At the end of the quiz, based on your
answers score on 10 would be assigned
answers a score on 10 would be assigned
• We will randomly select the person who gets
th fi
the first question and keep going in a roundrobin fashion (will explain this in person).
• In TRUE/FALSE questions an explanation is
needed. Question 1
• If X is the sum of two i.i.d. random variables
is the sum of two
random variables
X1 and X2, i.e. X = X1+X2, then X can also be
written as
written as X = X1+X1 = 2X1. True or False?
2X True or False? Question 2
• If X and Y are random variables with the
and are random variables with the
same mean and standard deviation, then the
CDF must be the same for both True or
CDF must be the same for both. True or
False? Question 3
• In the recent repairshop problem where X2
the recent repair
problem where
depends on X1 so that E[X2X1] = 4X1 and
Std
Std.Dev.[X2X1] = X1, is X1 independent of
is
of
X2? In other words, is E[X1X2]=E[X1]? Question 4
• A r.v. X has PDF f(x) = (1x)/2 for 1 < x < 1
has PDF f(x) (1
for
and f(x) = 0 otherwise. What is E[X]? Question 5
•
•
•
• F(x) = 0
if
if x < 0
F(x) = sin(x) if 0 <= x <= /2
F(x) = 1
if x > /2
Can the above F(x) be a CDF of a r.v.? Question 6
• If a continuous r.v. X has standard deviation
continuous
has standard deviation
equal to the mean, then X is exponentially
distributed True or False?
distributed. True or False? Question 7
• All the examples of random quantities we
the examples of random quantities we
have seen in class are either absolutely
discrete or absolutely continuous Give an
discrete or absolutely continuous. Give an
example of a random quantity that is partially
discrete and partially continuous
discrete and partially continuous. Question 8
• If X is a positivevalued continuous r.v., i.e.
is positive
continuous
F(0) = 0, then E[ X ] [1 F ( x)]dx
0 • True or False? Question 9
• There are two paths to go from A to B. One
are two paths to go from to
One
takes exp(1/3) time and the other takes
Unif(0 time If you want to make it before
Unif(0,6) time. If you want to make it before 3
time units from A to B which path would you
choose?
choose? Question 10
• Can you have two positivevalued continuous
you have two positive
continuous
random variables X and Y such that the PDF
of one is always greater than that of the
of one is always greater than that of the
other? Question 11
• Can you have two positivevalued continuous
you have two positive
continuous
random variables X and Y such that the CDF
of one is never less than that of the other?
of one is never less than that of the other? Question 12
• A doubleexponential r.v. has PDF for all x as
double
has PDF for all as f ( x) • What is F(0)?
is F(0)? 2 e  x Question 13
• For a CDF
CDF x0
0 ax b 0 x 1 F ( x) cx 0.5 1 x 2 1
x2 • Compute a, b and c Question 14
• If 25 samples are drawn according to an
25 samples are drawn according to an
exponential distribution with mean 1 and the
25 numbers are sorted then what would you
25 numbers are sorted, then what would you
expect the 13th number to be in this sorted
list? In other words what is the median of an
list? In other words, what is the median of an
exponential random variable? Question 15
• Let X ~ exp(), and for some constant a
exp( and for some constant
(such that a>0), Y = aX
• True or false? Y ~ exp(/a)
or false?
exp( Question 16
• X1, X2, …, Xn are i.i.d. r.v. with mean and
with mean
standard deviation .
• X = (X1+X2+…+Xn)/n
(X
• What are E[X] and Var[X]? Question 17
• For a positivevalued r.v. X, the coefficient of
positive
the coefficient of
variation is defined as Var[ X ]
COV ( X ) E[ X ]
• If X is Erlang, then COV(X) <= 1 and if X is
exponential COV(X)=1. True or False? Question 18
• True or false? Var[X] can be expressed as
or false? Var[X] can be expressed as Var[ X ] E { X E[ X ]} 2 Question 19
• The mode of a continuous r.v. X is the value
mode of continuous
is the value
of x for which the PDF f(x) attains the
maximum value What is the mode of an
maximum value. What is the mode of an
Erlang(2,) r.v.? Question 20
• If X ~ exp(3), what is P{X>3}?
exp(3) what is P{X>3}? Question 21
• If X ~ unif(a,b), what is P{X<(a+b)/2}?
unif(a
what is P{X<(a+b)/2}? Question 22
• If X ~ Erlang(k,), what is E[X2] ?
Erlang(k
what is E[X Question 23
• If X ~ Normal(0,1), and P{X>3} = a, then
Normal(0
and P{X>3}
then
what is P{3 < X < 3} in terms of a? Question 24
• If X and Y are iid Erlang(3,) r.v.’s what is
and are iid Erlang(3
what is
E[X+Y] ? Question 25
• If X ~ Unif(4,10), what is the standard
Unif
what is the standard
deviation of X? ...
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 Spring '08
 Klutke

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