R help - Jan 2009 - generic questions about probability and simulation -- not directly related to R

Dear helpers,

As the title says, my question is not directly related to R.
I find, however, that there are many people who are both knowledgeable and
kind in this email list, and so decided to give it a try.

I do stochastic simulations.  Parameter values used in simulation often come
from the observations of the real word phenomena.
Parameter values are often given as "rates" (of change),
"time", or
"probabilities".
I am confused about how I go about converting parameters given with
different units.

For example, I have a discrete time Markov model that describes the
following process:

A -> B -> C

Let's suppose that I am given average time that individuals stay at A,
"dA",
as 3 days.  We assume that "dA" is exponentially distributed.
Similarly, "dB" follows an exponential distribution with average 1000
days.


I decide to simulate the model with a time step corresponding to one day.

Would any of the following be correct?
a. A probability an individual makes transitions from A to B is 1/3.
Likewise, transition from B to C occurs with probability 1/1000.
b. If I reduce the size of time step as 0.1 day, then the transition
probability from A to B is 0.1*(1/3).  Likewise,  transition probability
from B to C is 0.1*(1/1000)
c. The size of time step must not be larger than 3 day, which makes the
transition probability to 1.
d. if parameters values are given rates of change, then I can directly
translate them to a probabilities per unit time.  There is no difference
between a rate and probability per time.

How do we know about the reasonable size of time steps?

Any help would be greatly appreciated.  Also, could anybody suggest pointers
or books that can be useful in this regard?

Sincerely,

-- JH

	[[alternative HTML version deleted]]

If the time until change is exponentially distributed with a mean of 3, then the
probability of changing in the first day is:
> pexp(1,1/3)
[1] 0.2834687

The same idea will work for all the other statements below (none of which are
true) including for time steps greater than 3 days.

Hope this helps,

-- 
Gregory (Greg) L. Snow Ph.D.
Statistical Data Center
Intermountain Healthcare
greg.snow at imail.org
801.408.8111

> -----Original Message-----
> From: r-help-bounces at r-project.org [mailto:r-help-bounces at r-
> project.org] On Behalf Of Jong-Hoon Kim
> Sent: Sunday, January 25, 2009 6:58 PM
> To: r-help at r-project.org
> Subject: [R] generic questions about probability and simulation -- not
> directly related to R
> 
> Dear helpers,
> 
> As the title says, my question is not directly related to R.
> I find, however, that there are many people who are both knowledgeable
> and
> kind in this email list, and so decided to give it a try.
> 
> I do stochastic simulations.  Parameter values used in simulation often
> come
> from the observations of the real word phenomena.
> Parameter values are often given as "rates" (of change),
"time", or
> "probabilities".
> I am confused about how I go about converting parameters given with
> different units.
> 
> For example, I have a discrete time Markov model that describes the
> following process:
> 
> A -> B -> C
> 
> Let's suppose that I am given average time that individuals stay at A,
> "dA",
> as 3 days.  We assume that "dA" is exponentially distributed.
> Similarly, "dB" follows an exponential distribution with average
1000
> days.
> 
> 
> I decide to simulate the model with a time step corresponding to one
> day.
> 
> Would any of the following be correct?
> a. A probability an individual makes transitions from A to B is 1/3.
> Likewise, transition from B to C occurs with probability 1/1000.
> b. If I reduce the size of time step as 0.1 day, then the transition
> probability from A to B is 0.1*(1/3).  Likewise,  transition
> probability
> from B to C is 0.1*(1/1000)
> c. The size of time step must not be larger than 3 day, which makes the
> transition probability to 1.
> d. if parameters values are given rates of change, then I can directly
> translate them to a probabilities per unit time.  There is no
> difference
> between a rate and probability per time.
> 
> How do we know about the reasonable size of time steps?
> 
> Any help would be greatly appreciated.  Also, could anybody suggest
> pointers
> or books that can be useful in this regard?
> 
> Sincerely,
> 
> -- JH
> 
> 	[[alternative HTML version deleted]]
> 
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