Help with derivation of probability density of {event generation} & {event detection}

#1
I would like to specify a new probability distribution that relates to an event of size [TEX] M [/TEX] being produced by some process and subsequently detected.

Some assumptions :

1) If the event is detected there is no loss of information regarding its size.

2) There are no false detections - An event detected that has not been produced in reality.

3) Probability of Detection only depends on the size of the observation.

I have tried many methods of deriving such a distribution from the two believed constituents (generation & detection), but each time I'm not satisfied with the form of the resulting distribution.

My most recent attempt has been to derive the density of [TEX] M^*[/TEX] (the size of an observation that has also been detected) accordingly :

[TEX] f_{M^*}(m) = \lim_{h \to 0} \frac{P[{\lbrace m < M \le m \text{ \plus } h \rbrace} \bigcap {\lbrace D(m, m \text{ \plus } h) = 1 \rbrace}]}{h} [/TEX]

where [TEX] M [/TEX] is the RV representing the size of the generated event & [TEX] D [/TEX] the RV representing the detection of the event :

(* Please excuse this part - I'm having trouble with the \cases keyword)

[TEX] D(a,b) = \lbrace \matrix{1 \text{ if the event in the interval [a,b] is detected} \cr 0 \text{ if the event in the interval [a,b] is NOT detected}} [/TEX]

where a < b

If the detection and generating events are independent then the numerator can be factored into the two components. My problem then is that when I compute the limit, the numerator approaches 0 too rapidly and the density vanishes.

Any pointers on this matter would be appreciated - if another approach is more desirable as well.

Thanks in advance
CJDW
 

Dason

Ambassador to the humans
#2
Re: Help with derivation of probability density of {event generation} & {event detect

The [noparse][/noparse] tags are more reliable than the TEX tags here if you want to give those a try instead.
 

Dason

Ambassador to the humans
#4
Re: Help with derivation of probability density of {event generation} & {event detect

Do you know the marginal distribution of the sizes? Do you have a specific form for the probability of detection given the size of the object?
 
#5
Re: Help with derivation of probability density of {event generation} & {event detect

Yes, I have specifications for both distributions. The event size distribution is a truncated exponential (between \(m_{min}\) and \(m_{max}\)), therefore with CDF :

\( F_M(m) = \frac{1 - e^{-\beta (m - m_{min})}}{1-e^{-\beta(m_{max} - m_{min})}} \)

For the moment, until I get the concept straightened out, I have simply used

\( f_D(m) = \frac{(\lambda + 1) m^\lambda}{m_{max}^{\lambda + 1} - m_{min}^{\lambda + 1}} \)

and therefore

\( F_D(m) = \frac{m^{\lambda+1} - m_{min}^{\lambda + 1}}{m_{max}^{\lambda + 1} - m_{min}^{\lambda + 1}} \)

When appropriately plugging these into the limit in the original post, I end up with an expression containing the factor :

\(\lim_{h \to 0}{\frac{[(m+h)^{\lambda + 1} - m^{\lambda + 1}][e^{-\beta h} - 1]}{h}}\)

When computing this limit it becomes 0 and !poof! vanishes my density.

Perhaps I am going down the wrong path, but what I am looking to achieve is to obtain a random variable

\( M^* = M ~ * ~ D \) with D as in the original post.

PS: @Dason : Thanks - the \( tags work superbly. :) :tup: :)\)
 

BGM

TS Contributor
#6
Re: Help with derivation of probability density of {event generation} & {event detect

Seemingly you mean that the distribution of the size random variable is known (as a truncated exponential) and the conditional probability of detection given size is also known. Now you want to determine the conditional distribution of size given detection. Therefore you simply need the Bayes Theorem?
 
#7
Re: Help with derivation of probability density of {event generation} & {event detect

...and the conditional probability of detection given size is also known. Now you want to determine the conditional distribution of size given detection. Therefore you simply need the Bayes Theorem?
I do not know how to thank you BGM. Through your formulation of my problem I have resolved so many issues. \( f_{D|M} \) and \( f_{M|D} \) (however basic), made the WORLD's difference.

My very first route and method (not explained in this thread), experimentally always worked when simulating from the distribution, but the rigor was missing, and so I always doubted the soundness of my approach. But thanks to your nudge everything worked out PERFECTLY.

Once again, thank you very much.