Problem Set # 1
(Due Wednesday, September 22 at beginning of lecture)
Economics 709
Fall 2010
As mentioned in the rst class, you may work in groups. However, you must write up your own solutions in your own words. CB refers to Casella and Berger. 1. CB 1.
Problem Set 1 (Due 9/19/2011 in class)
September 11, 2011
1. Isaac Newton and Samuel Pepys debated the following question: is the probability of
tossing at least one six in six tosses with a fair die smaller than, equal to, or larger
than the probability
Fall 2011
Econ 709
Prof. Xiaoxia Shi
Problem Set #4 Solutions
TA: SeoJeong (Jay) Lee ([email protected])
1.
Since g () is one-to-one, the inverse g 1 () is well-dened. Assume g 1 (z ) is dierentiable. Let
W = X . Then (f (X ), g (Y ) = (W, Z ) where f () i
Problem Set 4 (Due 10/26/2011 in class)
October 16, 2011
1. The random variable X conditional on another random variable Y has pdf: fX |Y (x|y ) y :
fY (y ) > 0. Let Z = g (Y ) for a one-to-one function g . Show that fX |Y (x|y ) =
fX |Z (x|g (y ).
2. 2.1
Fall 2011
Econ 709
Prof. Xiaoxia Shi
Problem Set #3 Solutions
TA: SeoJeong (Jay) Lee ([email protected])
1.
(a) Since EX 2 < , M SE (a) = E (X a)2 = EX 2 2aEX + a2 . The rst-order condition (FOC)
is
dM SE (a)
= 2EX + 2a = 0,
da
and the second-order sucient
Problem Set 3 (Due 10/17/2011 in class)
October 5, 2011
1. For a random variable X with nite second moment, let a scalar a be a guess of X s
value.
(a) Find the a that minimizes the mean squared error: M SE (a) = E (X a)2 . What
is the minimum mean-square
Fall 2011
Econ 709
Prof. Xiaoxia Shi
Problem Set #2 Solutions
TA: SeoJeong (Jay) Lee ([email protected])
1.
(simulation exercise) Omitted.
2.
(a) Let X and Y be random variables for the number of sixes Player 1 and 2 gets, respectively. Let
T be a random v
Problem Set 1 (Due 10/3/2011 in class)
September 21, 2011
1. Provide simulation evidence for the Monty Hall problem. Any programming language
is acceptable. Hand in code with clear description of every line and a (or more)
paragraph at the beginning descr
d zVQ 56Q C C o a9 m8a9 ( gmvQI @vcQI r9 P a9 E U E A U o ` EC EC w U o ` @ecI E 44 W4 $4W U ` f p2wzDQvcgI E 4pr~B~V0h!QymvA mvcgI E mvA mvcQI "4q~ U E ` f U E ` f A 4 q )&)4 $ 4 " 4 Byh6%Vy0)W )q & ~ |4 W q 4 ) )& ) A " ) " & U ` f A )& ) 4) 0'r~5qv)hm
Fall 2011
Econ 709
Problem Set #1 Solutions
TA: SeoJeong Jay Lee ([email protected])
1.
Let A and B be events tossing at least one six in six tosses, tossing at least two sixes in twelve
tosses, respectively. We calculate and compare P (A) and P (B ). Then