w******l
发帖数: 58 | 1 A company has a value V which is uniformly distributed between 0 and 1. you
are planning to place a bid B for the company. If B is smaller than V, then
your bid loses and you get nothing; if B is larger than V, you get to purcha
se the company at price B, and the company will end up being worth 1.5 * V.
What price B should you bid to maximize your profit?
On a sheet of paper, you have 100 statements written down. the first says, "
at most 0 of these 100 statements are true." the second says, "at | s*******r
发帖数: 63 | 2 1. Suppose we bid x, x\in[0,1], the profit can be given by
P = (1.5V-x)*\delta_{x-V},
where \delta_{x-V} is the unit step function, \delta(x)=1 for x>=0, 0 for x<
0. Here, note that x is a scalar and V is a random variable.
we want to max E[P]=E[(1.5V-x)*\delta_{x-V}]=-x^2/4, thus x=0. shouldn't bid
anything.
PS:
E[(1.5V-x)*\delta_{x-V}] = \int_0^1 (1.5V-x)*\delta_{x-V} dV
= \int_0^x (1.5V-x) dV
= -x^2/4
2. suppose there are x statements are true, then 1st,...,nth statements are
false and the re | w******l
发帖数: 58 | 3 yeah, that's also what I got.
I was a little surprised to see that
we should not bid anything. But it makes
sense after some thinking.
x<
bid
【在 s*******r 的大作中提到】
: 1. Suppose we bid x, x\in[0,1], the profit can be given by
: P = (1.5V-x)*\delta_{x-V},
: where \delta_{x-V} is the unit step function, \delta(x)=1 for x>=0, 0 for x<
: 0. Here, note that x is a scalar and V is a random variable.
: we want to max E[P]=E[(1.5V-x)*\delta_{x-V}]=-x^2/4, thus x=0. shouldn't bid
: anything.
: PS:
: E[(1.5V-x)*\delta_{x-V}] = \int_0^1 (1.5V-x)*\delta_{x-V} dV
: = \int_0^x (1.5V-x) dV
: = -x^2/4
| L****a
发帖数: 572 | 4 qustion 1, B=0
question 2, answer = 50 | j***4
发帖数: 3119 | 5 B=0 can't be right for question 1. what's your final profit? 0?
the E(1.5V-x) assumption is based on you get the purchase, which implies X>=
V. So the bid should be X=V. | o****e
发帖数: 80 | 6 i think 99 statements are true, the only false is the first one
any thoughts?
you
then
purcha
.
"
of
10
a
【在 w******l 的大作中提到】
: A company has a value V which is uniformly distributed between 0 and 1. you
: are planning to place a bid B for the company. If B is smaller than V, then
: your bid loses and you get nothing; if B is larger than V, you get to purcha
: se the company at price B, and the company will end up being worth 1.5 * V.
: What price B should you bid to maximize your profit?
: On a sheet of paper, you have 100 statements written down. the first says, "
: at most 0 of these 100 statements are true." the second says, "at
| l******8
发帖数: 32 | 7 1. Max E[pi] = E[1.5v-B | V < B]*B = -1/4*B^2 --> B = 0
2. If statement i is true --> all the statements after are true. So
n-i+1 = i-1 --> i = 51.
for x<
bid
【在 s*******r 的大作中提到】
: 1. Suppose we bid x, x\in[0,1], the profit can be given by
: P = (1.5V-x)*\delta_{x-V},
: where \delta_{x-V} is the unit step function, \delta(x)=1 for x>=0, 0 for x<
: 0. Here, note that x is a scalar and V is a random variable.
: we want to max E[P]=E[(1.5V-x)*\delta_{x-V}]=-x^2/4, thus x=0. shouldn't bid
: anything.
: PS:
: E[(1.5V-x)*\delta_{x-V}] = \int_0^1 (1.5V-x)*\delta_{x-V} dV
: = \int_0^x (1.5V-x) dV
: = -x^2/4
| p*****k
发帖数: 318 | 8 ogtree, if 99 statements are true, then all the statements
"at most x statements are true" with x<=98 are all false,
contradiction
btw, ive noticed that you have been digging out old posts.
if you have time, i would suggest you starting a thread with
a compilation of links to these problems. other ppl could
contribute too. it would be a very good thing to do | o****e
发帖数: 80 | 9 yes, i am reading the old articles.
i am not sure what do you mean about 'starting a thread with a compilation
of links to these problems.' please
be more specific. i would be glad to do it, others and i could be beneifit
from it. thank you.
【在 p*****k 的大作中提到】
: ogtree, if 99 statements are true, then all the statements
: "at most x statements are true" with x<=98 are all false,
: contradiction
: btw, ive noticed that you have been digging out old posts.
: if you have time, i would suggest you starting a thread with
: a compilation of links to these problems. other ppl could
: contribute too. it would be a very good thing to do
| p*****k
发帖数: 318 | 10 ogtree, this is great! many thanks for starting the thread
- i will try contributing soon
the format i had in mind is something like:
http://www.mitbbs.com/article_t/JobHunting/31505215.html
definitely low-tech (and takes lots of time, so greatly
appreciate you doing this);
better format would be something like:
http://spellscroll.com/
but these two are mostly for IT jobs.
thought it would be more tailored to this board if all the
questions are from actual quant interviews | o****e
发帖数: 80 | 11 i see, i will try to follow the first format. and try to add actual
interview questions. the second one i don't
know how to build up website:( thank you very much for always being warm-
hearted on this board, i am sure
many ppl not only me have been benefited a lot from your elegant solutions.
【在 p*****k 的大作中提到】
: ogtree, this is great! many thanks for starting the thread
: - i will try contributing soon
: the format i had in mind is something like:
: http://www.mitbbs.com/article_t/JobHunting/31505215.html
: definitely low-tech (and takes lots of time, so greatly
: appreciate you doing this);
: better format would be something like:
: http://spellscroll.com/
: but these two are mostly for IT jobs.
: thought it would be more tailored to this board if all the
|
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