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Holland Kin.
===
Subject: Hard probability problem
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i8LLi8i01218;
Does anyone have any ideas on how to solve this:
If we have an empty room, and every week there is a .6
probability for
a person to join, .2 probabillity for a person to leave, and
.2
probability for no change (only one action can be performed
per week),
what is the probability that there are at least 40 people in
the room
after 104 weeks?
Mike
===
Subject: Re: Hard probability problem
X-RFC2646: Original
> Does anyone have any ideas on how to solve this:
> If we have an empty room, and every week there is a .6
probability for
> a person to join, .2 probabillity for a person to leave,
and .2
> probability for no change (only one action can be performed
per week),
> what is the probability that there are at least 40 people
in the room
> after 104 weeks?
> Mike
It's not quite true in the first week, Since no
one is there,
there can not
be
a .2 probability of someone leaving. Also I assume you mean
that .2
probability for a person to leave means a .2 probability that
anyone in the
room leaves - i.e. only one person can leave per week. Have
you studied
Markov
Chains and Random walks? This is the kind of area this falls
into.
Bill
===
Subject: Re: Hard probability problem
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i8M1Fdd19703;
Hi Bill,
I have not studied randoma walks and markov chain. I have
looked them
up on mathworld.wolfram, but that didn't give me much info.
Regarding the first week, you can assume that there are 2
people in
the room.
Could you maybe give me some clues on how I should start
thinking
about this problem?
I appreciate your help.
Mike
> Does anyone have any ideas on how to solve this:
> If we have an empty room, and every week there is a .6
probability
for
> a person to join, .2 probabillity for a person to leave,
and .2
> probability for no change (only one action can be
performed per
week),
> what is the probability that there are at least 40 people
in the
room
> after 104 weeks?
> Mike
>It's not quite true in the first week, Since no
one is there,
there
can not be
>a .2 probability of someone leaving. Also I assume you mean
that .2
>probability for a person to leave means a .2 probability
that anyone
in the
>room leaves - i.e. only one person can leave per week. Have
you
studied Markov
>Chains and Random walks? This is the kind of area this falls
into.
>Bill
===
Subject: Re: Hard probability problem
X-RFC2646: Original
> Hi Bill,
> I have not studied randoma walks and markov chain. I have
looked them
> up on mathworld.wolfram, but that didn't give me much
info.
> Regarding the first week, you can assume that there are 2
people in
> the room.
> Could you maybe give me some clues on how I should start
thinking
> about this problem?
> I appreciate your help.
You changed the initial problem. Is this a homework problem
or something you
are doing on your own? In any case, I really can't give you
much additional
help because that is what this is - random walk/Markov chain.
If this is
something you are studying on your own, you might research
discrete Markov
chains and transition matrices. If this is homework, you
might give me some
suggestions as to what areas you are studying now and that
might trigger
something. But I would not count on it. :)
Or perhaps someone else has a different approach.
Bill
> Mike
> Does anyone have any ideas on how to solve this:
> If we have an empty room, and every week there is a .6
probability
> for
> a person to join, .2 probabillity for a person to leave,
and .2
> probability for no change (only one action can be
performed per
> week),
> what is the probability that there are at least 40 people
in the
> room
> after 104 weeks?
> Mike
>It's not quite true in the first week, Since no
one is
there, there
> can not be
>a .2 probability of someone leaving. Also I assume you mean
that .2
>probability for a person to leave means a .2 probability
that anyone
> in the
>room leaves - i.e. only one person can leave per week. Have
you
> studied Markov
>Chains and Random walks? This is the kind of area this
falls into.
>Bill
===
Subject: Re: Hard probability problem
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i8M1Fdq19713;
Hi Bill,
I have not studied random walks and markov chain. I looked
them up
online at mathworld and did not find much there.
Could you maybe give me some clues on how i should start
thinking this
problem?
Mike
> Does anyone have any ideas on how to solve this:
> If we have an empty room, and every week there is a .6
probability
for
> a person to join, .2 probabillity for a person to leave,
and .2
> probability for no change (only one action can be
performed per
week),
> what is the probability that there are at least 40 people
in the
room
> after 104 weeks?
> Mike
>It's not quite true in the first week, Since no
one is there,
there
can not be
>a .2 probability of someone leaving. Also I assume you mean
that .2
>probability for a person to leave means a .2 probability
that anyone
in the
>room leaves - i.e. only one person can leave per week. Have
you
studied Markov
>Chains and Random walks? This is the kind of area this falls
into.
>Bill
===
Subject: Re: Solving for the n root.
>a^n = 2a+n+1
>n^a = 2n+a+1
>n>a
>
>How do you solve and what is the solution?
> n = 3, a = 2. This was solved by inspection, really, not
by
> any systematic analysis.
> Another integer solution is n = 1, a = -2. I suspect that
it may be the
> only other real solution.
Perhaps it would interesting to note that if the inequality
is weakened to
n >= a, we get two more real solutions: n = a = 2.40199262...
(obtained
by solving x^x = 3x + 1 numerically) and n = a = 0 (assuming
that 0^0 is
taken to be 1, as is often done).
David Cantrell
> One question is: what are a, n? Integers, real numbers,
or what?
> I expect that the solution above is the unique real
solution in
> any case, but it's probably not hard in the integer case
to put
> small bounds on the size of a and n (e.g. |a|, |n| < 4)
and then
> since this leaves only finitely many possibilities, one
can argue
> by process of elimination if nothing else. In the real
case,
> something more sophisticated may be required to
demonstrate
> uniqueness. It looks messy.
===
Subject: problem about differentiation of matrix
Given a 1xn matrix A and nx1 matrix B, what is
d(AB)^2/dA ?
Can anyone give me some cue?
===
Subject: Re: problem about differentiation of matrix
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i8MFw4q00967;
>Given a 1xn matrix A and nx1 matrix B, what is
>d(AB)^2/dA ?
>Can anyone give me some cue?
Ignore the fact that A and B are matrices and just do the
derivative as if they were numbers. What is the derivative of
(xy)^2 with respect to x?
In any algebra in which the derivative can be defined, the
product
rule and chain rule are true.
===
Subject: Re: For whose want to know there level of knowledges
in analisis
===
Subject: Re: For whose want to know there level of knowledges
in analisis
> For continuous(only) function in(a,b) every point
> from (a,b) is a local extremum.Proof that f=const.
>If f isn't constant on (a, b), then there are c and d in
>(a, b) such that c < d, and either f(c) < f(d), or f(c) >
>f(d). Assume that f(c) < f(d); the other case is similar
>(or if f(c) > f(d), you can look at the function -f instead).
>Define g(x) = max{f(y) : y in [c, d]}. Note that g is a
>non-decreasing function on [c, d], f(x) <= g(x) for all x in
>[c, d], g(c) = f(c), g(d) = f(d), and g is continuous on [c,
d].
g is a constant function with, I presume, domain (a,b).
>If f has a local maximum at some u in (c, d), then there is
a v > u
>such that g is constant on [u, v]. If f has a local minimum
at
>some u in (c, d), then there is a v < u such that g is
constant on
>[v, u].
g is constant on all of (a,b).
>Let U be the union of all open intervals in [c, d] on which
>g is constant; the last two sentences of the previous
>paragraph show that U is dense in [c, d] (i.e., every point
>of [c, d] is a limit point of U). Define an equivalence
>relation, ~, on U: for x, y in U with x < y, x ~ y if and
>only if g is constant on [x, y]. The equivalence classes of
>this relation are pairwise disjoint open intervals, and of
>course g is constant on each of them; let E be the set of
>these equivalence classes.
>Each of these equivalence classes must contain a rational
>number, so there can be only countably many equivalence
>classes. But g is continuous, so the range of g contains
>the interval [g(c), g(d)], which is uncountable.
Codomain g, I presume, is R. Range of g is a single point.
>Pick a y in [g(c), g(d)] that is not the value of g on any
>member of E; there must be some x in [c, d] such that g(x) =
y.
>Clearly x is not in any member of E; indeed, if V is in E,
>then x cannot even be an endpoint of V. But this
>contradicts the fact, previously established, that g is
>constant on a closed interval having x as one endpoint.
Would you clarify how is g defined?
----
===
Subject: Re: For whose want to know there level of knowledges
in analisis
===
> Subject: Re: For whose want to know there level of
knowledges in
analisis
> For continuous(only) function in(a,b) every point
> from (a,b) is a local extremum.Proof that f=const.
>If f isn't constant on (a, b), then there are c and d in
>(a, b) such that c < d, and either f(c) < f(d), or f(c) >
>f(d). Assume that f(c) < f(d); the other case is similar
>(or if f(c) > f(d), you can look at the function -f
instead).
>Define g(x) = max{f(y) : y in [c, d]}. Note that g is a
>non-decreasing function on [c, d], f(x) <= g(x) for all x
in
>[c, d], g(c) = f(c), g(d) = f(d), and g is continuous on
[c, d].
> g is a constant function with, I presume, domain (a,b).
Obviously that should have been g(x) = max{f(y) : y in [c,
x]}.
[...]
Brian
===
Subject: Re: i need a riddle solved!!!
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i8MBiv906363;
Riddle:
Presto...Abracadabra...First one then the other.
I'm _ _ _ _ _ _ _ _ _ _ _ _ _ _
===
Subject: Re: i need a riddle solved!!!
> Riddle:
> Presto...Abracadabra...First one then the other.
> I'm _ _ _ _ _ _ _ _ _ _ _ _ _ _
o f f t o p i c p o s t e r
--
Wayne Brown (HPCC #1104) | When your tail's in a crack, you
improvise
fwbrown@bellsouth.net | if you're good enough. Otherwise you
give
| your pelt to the trapper.
e^(i*pi) = -1 -- Euler | -- John Myers Myers,
Silverlock
===
Subject: Discrete Mathematical structures
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i8ME4cO21612;
CAN ANYONE HELP TO THESE HOMEWORK QUESTIONS...
1) A = B = R A relation R: A .b3_ B defined by R(a) = ea
Determine whether that relation as defined is also a function;
and if
so, what is the range of the function?
2) A = R B = Z a function f is defined f: A .b3_ B so that
f(a)
= .b3.b2 a .b3 called the
ßoor function. Find the
images produced by
the following inputs:
a) f( 3.67) = ____________
b) f( -3.67) = ____________
c) f( - .b3 21) = __________
d) f( .b3 17 ) = ____________
3) A = { 1,2,3,4} B = { a,b,c,d} f : A .b3_ B f =
{ (1,a), (2,a), (3,c), (4,d) }
Is this function; one-to-one, onto, both; or neither?
Give supporting evidence for your decisions.
4) Using the same function and sets from part three now
determine
the inverse of f.
Is f V1 also a function? Explain your choice of
answer.
5) Let set A be a finite set of four elements ; A = { a,b,c,d}
define
A* as the set of all finite sequences of elements taken from
A. Let l
: A* .b3_ Z be a function that matches each string w with the
integer
number of elements in that string. This function l(w) has
outputs of
the length of those strings.
a) Verify the l(w) is everywhere defined on A*.
b) Determine whether l(w) is onto or not and give supporting
evidence.
c) Prove that l(w) is not one-to-one.
6) Show that f( n ) = n100 , is O(g) for the function g(n) =
2n ;
but that
f(n) is not g.
7) Order the following functions by -classes from
lowest or slowest
growing to fastest growing.
F1 (n) = 5 n lg (n)
F2 (n) = 6 n2 + 4n V6
F3 (n) = 2.5n
F4 (n) = lg(n4)
F5 (n) = 12,271
F6 (n) = -31n
F7 (n) = n !
F8 (n) = .b3 n { 5n
8) Which of these following functions are permutations of R ?
f : R .b3_ R
a) f(a) = a3
b) f(a) = e a
c) f(a) = 2a + 5
For problems 9 - 12 we will use the set A = { 1,2,3,4,5} and
the four
permutations of A
P1 = .b3 1 2 3 4 5 .b3.83 P2 = .b3 1
2 3 4 5 .b3.83 P3 = .b3 1 2 3 4 5 .b3.83 P4 =
.b3 1 2 3 4 5 .b3.83
.b3 4 5 2 1 3 .b3
.b3 3 1 2 4 5 .b3 .b3 5
4 3 2 1
.b3 .b3 4 3 1 2 5 .b3
9) Compute: a) P1-1 = _______________
b) P3 o P2 = _______________
c) P2-1 o P2 = _________________
10) How many permutations of set A are possible? How many
even?
How many odd?
11) a) Write P3 as a product of disjoint cycles:
b) Write P2 as a product of transpositions:
c) Determine which of our four given permutations are even
and which
are odd:
12) Compute these products :
a) ( 1 , 5, 3) o ( 2, 4) =
b) ( 2, 5, 4 ) o ( 1 , 3) =
c) (1, 4) o ( 3, 5) =
13) If we assume that 10,370 account records need to be
stored using a
hashing function h ; which takes the first three digits of an
account
number as one value and the last four as a second value adds
these two
and applies the Mod- 93 function to find a location list.
a) How many linked lists will we need ?
b) If an approximately even distribution is achieved, roughly
how many
records will be stored in each lsit?
c) Compute the location number for the following accounts;
4738124
1253135
3086291
14) Prove rule #3 (na) is lower than
(nb) if and only if
0The weather forecastor reports that the porbability of rain
tomorrow
>is 10%.
>How do you interpret the probability?
> What that really means is Looking over our data bank of past
> weather, we find that, of all days when our various
indications
> (temperature, air pressure, wind speed, weather upwind,
etc) we the
> same as now, it rained on 10% of those days.
Another variable is the size of the area covered by the
forecast: do
they count it as an occurrence of rain if there was rain
anywhere in
that area? What if it rained for only 3 seconds? It doesn't
seem like a
well-defined probability.
--
john