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

I would like to specify a new probability distribution that relates to an event of size M 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 $M^*$ (the size of an observation that has also been detected) accordingly :

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

where $M$ is the RV representing the size of the generated event & $D$ the RV representing the detection of the event :

$$D(a,b) = \cases{1 & \text{if the event in the interval [a,b] is detected} \cr 0 & \text{if the event in the interval [a,b] is NOT detected}}$$

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. So I believe the error is in my approach(!?!)

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

Thanks in advance $\\$ CJ de Witt

-