« first day (404 days earlier)
← previous day
next day →
last day (1071 days later) »
BAYMAX
1:11 PM
@SiongThyeGoh Hi!! can we discuss about this problem -
Siong Thye Goh
1:27 PM
hi
BAYMAX
Hi@SiongThyeGoh
How are u doing!!
Siong Thye Goh
i'm good
BAYMAX
I had a course on stochastic ut how do i think of this problm ny idea
Siong Thye Goh
perhaps induction?
BAYMAX
also what is the motivation behind hititng time
any motivation to think about this question
Siong Thye Goh
1:31 PM
motivation behind hitting time?
hmm
ah, first visiting time
BAYMAX
actually this was from our Masters exam previous year, i was just looking through a coupe topics as i hv to tutor students on this
Siong Thye Goh
it's like we want to know when do we go bankrupt
BAYMAX
oh my
oh gotcha
that is where we add the term Inf
Siong Thye Goh
oh when do we retire as we are having too much money if u r the positive thinking type of person
BAYMAX
haha :)
so that is denoted y $T_{0}$
Siong Thye Goh
1:36 PM
for this particular question
i'm not sure if the notation si standardized
BAYMAX
yes for this question
and values of $S_{0}$ increase by 1 with probability 1
sorry with probability p
and decrease by 1 with probability 1-p
Siong Thye Goh
means value of $S_{n+1} = S_n +1$ with probability $p$.
BAYMAX
ohh
and
Siong Thye Goh
$S_n$ means state at time $n$
BAYMAX
$S_{n+1} =S_{n} -1$ with probability $q=1-p$
Siong Thye Goh
1:41 PM
yup
BAYMAX
I see
Now we can think of the problem 1
$E(T_{0} | S_{0}=i) = i E(T_{0}|S_{0} =1)$
as u saif
said
by induction?
Siong Thye Goh
i dunno if that's the best method
but i think it should work
BAYMAX
so which method u would reccommend?
Siong Thye Goh
since i have no idea for now i would do induction?
=D
BAYMAX
ok
:)
I think how to show $E(T_{0}|S_{0}=2) = 2E(T_{0}|S_{0}=1)$
that means that time $t=0$
we start with initial state 2, then i m thinking what to write T0 as infimum
?
Siong Thye Goh
1:51 PM
maybe i have misguided u
not sure if induction works
maybe we needed independence
( i m thinking on the spot without solution in mind btw)
BAYMAX
oh ok ok
Siong Thye Goh
$E[T_0|S_0=2] = E[T_1|S_0=2] + E[T_0|S_1=1]$
seems to be still wrong
$E[T_0|S_0=2] = E[T_1|S_0=2] + E[T_0|S_0=1]$
BAYMAX
hm
Siong Thye Goh
we start from state $2$, to reach state $0$, we have to go through state $1$.
BAYMAX
2:08 PM
yes like markov
chain
Siong Thye Goh
i think it's a markov chain
BAYMAX
yes just we need to write
put into equations
perhaps we need t owrite in sigma notations
Siong Thye Goh
u understand the case where we start from $2$?
BAYMAX
2 o 1 to 0
Siong Thye Goh
yup
for general $n$, you have to move to $n-1$ first
then use induction
BAYMAX
2:20 PM
yes
but how t odo them in eqns
Siong Thye Goh
$$E[T_0|S_0=n] = E[T_{n-1}|S_0 = n] + E[T_{0}|S_0={n-1}]$$
then use induction hypothesis
BAYMAX
why $T_{n-1}$
Siong Thye Goh
i need to go over to $n-1$ first right
BAYMAX
from n to n-1
if going to $n-1$ is $T_{n}$
then where is $n$ that is $T_{n}$
?
Siong Thye Goh
??
hmm can't understand
BAYMAX
2:32 PM
I m not getting :'(
Siong Thye Goh
if u want to go from $n$ to $0$, we have to first pass through $n-1$ right
BAYMAX
yes
Siong Thye Goh
that amount of time is expected to be $E[T_{n-1}|S_0=n]$
then the time from $n-1$ to state $0$ would be $E[T_0|S_0=n-1]$
BAYMAX
oh ok its the notation trickss
yes ys got it
ok
so from $n$ to $0$
we go fro $n$ to $n-1$
then $n-1$ to $0$
then thsi is true
$E(T_{0}|S_{0}=n) = E(T_{n-1}|S_{0} = n) +E(T_{n-2}|S_{0}=n-1)+E(T_{0}|S_{0} = n-2)$
?
Siong Thye Goh
yes, but u could have just use induction to hide all the details
BAYMAX
2:39 PM
I expanded $E(T_{0}|S_{0}=n-1)$
using inductin
I still think how one could factor an $i$ out, its like adding the same thing i times
Siong Thye Goh
\begin{align}
oops
\begin{align}
E[T_0|S_0 = n) &= E[T_{n-1}|S_0=n] + E[T_0|S_0=n-1]\\
&= E[T_{n-1}|S_0=n] + (n-1)E[T_0|S_0=1]\\
&= E[T_{0}|S_0=1] + (n-1)E[T_0|S_0=1]\\
\end{align}
BAYMAX
how from first t osecond step?
like factoring n-1 out?
Siong Thye Goh
induction
write down the induction hypothesis and u have a tool to use
BAYMAX
it sems like we are using the result we want t o prove
Siong Thye Goh
we want to prove for case n, we assume it is true for n-1
BAYMAX
2:49 PM
Oh man yes
right right
Oh so we have it nice nice
$E(T_{0}|S_{0}=1)$
means the expected time
reuired for the wealth at 1 to reach 0
?
Why they would be different for two different cases like $p<q$ and $p \geq q$
Siong Thye Goh
i m guessing (i have not computed anything though i should know these results)
one of them is going to be infinity
and u can guess which one
=)
but i m uncertain as i have not done such things for years
BAYMAX
$p \geq q$
Siong Thye Goh
try to compute the value to find out?
BAYMAX
?
Siong Thye Goh
i m guessing but i m not sure
if one of them is infinity, it has to be that one
BAYMAX
2:55 PM
But if i use the result we obtained
$E(T_{0}|S_{0}=i) = i E(T_{0}|S_{0}=1)$
i just put $i=1$
to get the LHS =RHS
from state i to state 0
that is from state 1 t ostate zero
Siong Thye Goh
u have to think of how to use $p$ and $q$ somewhere
BAYMAX
$S_{1} = S_{0}+1$ with probability $p$
$S_{1} = S_{0} - 1$ with probability $q$
Siong Thye Goh
yes
u r moving in the right direction
BAYMAX
Hm i m not usre wht next
Siong Thye Goh
$$E[T_0|S_0=1]=q+pE[T_0|S_0=2]$$
BAYMAX
3:08 PM
Still trying :'(
what is the expression for p an d q?
Siong Thye Goh
take ur time
?
p is the probability to move to the right
q is the probability to move to the left right
i think i might have made a mistake
$$E[T_0|S_0=1] = q + p(1+E(T_0|S_0=2))$$
BAYMAX
$E(T_{0}|S_{0}=2) = E(T_{1}|S_{0}=2)+E(T_{0}|S_{0}=1)$
$E(T_{1}|S_{0}=2) = q$
as moving from state 2 to 1
Siong Thye Goh
actually the equation that i gave u... u just have to solve for $E[T_0|S_0=1]$
$E[T_1|S_0=2] = E[T_0|S_0 =1 ]$
BAYMAX
is it like this
we go from state 1 to 0
we can go thru 2 ways
1 t o0
Siong Thye Goh
u can directly go to $0$ or u can go to $2$ first
that is what the equation I wrote above means
BAYMAX
3:19 PM
yes
Siong Thye Goh
$$E[T_0|S_0=1] = q + p(1+E(T_0|S_0=2))$$
BAYMAX
1) going from state 1 to state 0
2) state 1 to stae 2 then stat 0
?
Siong Thye Goh
yup
BAYMAX
yes stae 1 to 0 is q
now from state 2 t ostate 0
?
why p is involved?
eventhough we are going left
that is 2 to 0
Siong Thye Goh
from state 1 to 0 take time 1 with probability q
BAYMAX
3:22 PM
yes
Siong Thye Goh
i mean with probability q, we go from state 1 to state 0, and that would take 1 unit
and with probability p, we go from state 1 to state 2 , and that would take ... the expression that i typed earlier
BAYMAX
state1 to stae2
is p
state 2 to state 0
?
Siong Thye Goh
the probability is p, and the time needed would be $1+E(T_0|S_0=2)$
we use the law of total expectation
*away a while
BAYMAX
wll be back soon
Siong Thye Goh
3:46 PM
back
BAYMAX
me too
oh ok
How about $p<q$, $p \geq q$
?
cases
Siong Thye Goh
the equation that i wrote
solve for $E[T_0|S_0=1]$
BAYMAX
like in question it has divided into cases of the above expectatio
Siong Thye Goh
solving that would give u a big hint
BAYMAX
q/1-2p
Siong Thye Goh
4:02 PM
hmmm
somehow i get $1/(1-2p)$
but yup, it's just algebra now
at least our denominator is the same
so now u can see what happens when $p<\frac12$ and what happens otherwise
BAYMAX
yes it must be $\frac{1}{1-2p}$
For $p=0.5$, it blows up
Siong Thye Goh
yup
BAYMAX
p<0.5, it simply is positive
p>0.5 its negative not possile
*possible
Siong Thye Goh
yup, it means it's also infinity
BAYMAX
oh ok
How we related p and q
q was independent here
Siong Thye Goh
4:10 PM
$p+q=1$ is n't it?
BAYMAX
yes
Siong Thye Goh
$p \ge q \iff p \ge \frac12$
BAYMAX
oh yes
Oh so our past first gor over
its deep no
Siong Thye Goh
erm... can't comprehend what was typed
BAYMAX
Like it has deep concepts involved
can u plz give solutions to lst two parts, i will try to understnad?
Siong Thye Goh
4:17 PM
the strategy is the same
induction
BAYMAX
its much time since i used stochastics
so i said
Here we are mulitplying
Siong Thye Goh
yes, it is by induction
induction is not relying on addition
BAYMAX
?
induction is not relying on addition?
$P(T_{0}<\infty|S_{0}=i)$
tells that probability to move from state $i$ to some finite state
Siong Thye Goh
use the idea that u ahve to go throught state i-1
if u understand the first part u should be able to do this as well
(whcih explians why only 1 mark)
BAYMAX
let me work on it
i will let u know
by ping
its night
hv t oeat hungry nd sleep
Siong Thye Goh
4:25 PM
eat more
BAYMAX
yeah
:)
Ok bye
i will discuss tmrw
Siong Thye Goh
bye
« first day (404 days earlier)
← previous day
next day →
last day (1071 days later) »
all rooms
Transcript for
Oct
15
Oct '18
26
Oct
27
Probability and Statistics
Any discussion on Probability and Statistics.For rendering LaT...
probability-theory
random-variables
statistics
join this room
about this room
00:00
06:00
12:00
18:00
all times are UTC
site design / logo © 2024 Stack Exchange Inc;
legal
mobile