Jointly Distributed Discrete Random Variables

[jingchyi87]

Hi all,

I am currently doing my Final Year Project on the topic of Optimal Placement of Suicide Bomber Detectors.
Given 2 dependent bomb detectors, I am trying to prove that the probability of detection in the intersected area will be larger than the individually covered areas, by working along the lines of jointly distributed discrete random variables and also joint probability mass functions.

Any suggestions on how to work on this? Thank you, and apologies if my question is not clear.

 

[Stephen Tashi]

You'll have to be more specific about the probability model that you are using for the detection. For example, if assume the detectors have an independent probability of detection, then you can show the probability of one or the other detecting is greater than the probability of one detecting. However the probabilistic nature of detection is due to the variety of bombs and methods of concealment encountered then two different detectors may both tend to succeed on one type of bomb and fail on the another type. So their probabilities of detection are not independent.

 

[jingchyi87]

hmmm... to put it simply, refer to the diagram:
1. X denotes the detector (both are of the same type)
2. the red arrow denotes the direction of travel of the threat

hence, the shaded red area is the matter of concern here... given that each detector has the same detection probability, does that intersection have an enhanced and increased detection probability?

i am utterly confused as to how to prove it mathematically using concepts of data/sensor fusion?

thank you

 

[Stephen Tashi]

As I said before, the answer depends on the probability model you are using for detection. If you don't have a probability model for detection then you'll never figure out what you're doing.

For example, what exactly is the event you are taking about? The probability of detection "at a given location" or the probability of detection "at a given time". If someone said "The probability of detection is 0.135", what would that mean? Is that the probability of detection every second? Or is the the probability of detection at anytime in the future when the target sits in one place?

What does the probability depend upon? Does it depend on the range of the detector to the threat? Does it depend on how long the target has been in the circle that defines the range of the detector?

 

[chiro]

From the OP's post I'm guessing that there is a binary model of some sort where there are two outcomes being detected and not-detected for each area.

Basically what you have is two indicator variables and depending on the situation they will be either depdendent or independent.

There are probability rules for the union of events as well as for intersection regarding individual probabilities.

For example we have P(A or B) = P(A) + P(B) - P(A and B) as well as independent events following P(A and B) = P(A)P(B) if two events are independent.

From what you have posted, you are interested in if using both detectors is better than using either one.

For this you need to consider P(A = a and B = b) as a joint distribution where A is an indicator variable and B is an indicator variable. It would help if you set up a table with your data to help us.

 

[jingchyi87]

firstly, apologies to Stephen for not clarifying your question beforehand, as you can see, my grasp of stats ain't exactly fantastic.

regarding to the probability model for detection, i am using an expression which depends on the length of path taken by the threat within the detector's range, extracted from Przemieniecki (2000) which goes smth like this:

Probability of detection P = 1 - e^-nl

where n is the detector's instantaneous detection rate, l is the length of the path taken within the detection radius...

with reference to chiro's post, it is correct in a sense. there are 4 different outcomes: false alarm, false clear, true alarm and true clear. but what i am interested to find out now is will the detection rate be better IF the 2 similar detectors used are dependent? sorry but there is no data, it's just a theorectical project...

its all kinda messy to me, i don't know how to start on this really, have been stuck for the past few weeks reading complicated journals and articles... basically the project scope is to come up with a mathematical model which minimizes casualty rates...

 

[chiro]

firstly, apologies to Stephen for not clarifying your question beforehand, as you can see, my grasp of stats ain't exactly fantastic.

regarding to the probability model for detection, i am using an expression which depends on the length of path taken by the threat within the detector's range, extracted from Przemieniecki (2000) which goes smth like this:

Probability of detection P = 1 - e^-nl

where n is the detector's instantaneous detection rate, l is the length of the path taken within the detection radius...

with reference to chiro's post, it is correct in a sense. there are 4 different outcomes: false alarm, false clear, true alarm and true clear. but what i am interested to find out now is will the detection rate be better IF the 2 similar detectors used are dependent? sorry but there is no data, it's just a theorectical project...

its all kinda messy to me, i don't know how to start on this really, have been stuck for the past few weeks reading complicated journals and articles... basically the project scope is to come up with a mathematical model which minimizes casualty rates...

Just to clarify, when you are talking about your two detectors, are you talking about having two physical detectors in different positions where the range of detection of both detectors overlap in some sense? Like say one detector on the left pointing to the right and one on the left pointing to the right?

If this is the case, do you want to want to find if the results of left detector are dependent on the results from right detector? Is this what you want to statistically determine?

 

[Stephen Tashi]

I sympathize with the problem that you have in researching the topic. Many papers that apply probability models to military operations research contain outright mathematical absurdities. Often such papers are only reviewed by people working for the military instead of referees from the general academic community.

The probability model you describe implies p(target detected at position x) and p(target detected at time t) can't be defined by inputs of position and time alone. It is dependent on the path that the target ("threat") took to reach position x at time t, not simply on x or t. By this theory if a target manages to reach position x at time t without being detected and remains stationary, then it can never be detected.

One problem with relating (1 - e^{-nl}) to an "instantaneous" detection probability is that the "instantaneous" detection rate must be defined with respect to a change in the path length. If one wishes to model that fact that a stationary target might eventually be detected then one must define an instantaneous detection rate as a function of time once the target has stopped.

The probability model also implies that if n > 0 that the if the target keeps moving within the detection range of the detector that it will eventually be detected with probability 1. What is n a function of? Are there targets for which n = 0 ?

The probabiliy model does not address the issue of whether two detectors of the same target have independent probabilities of detecting it or not.

You will have to consider the physics of the detection device and determine whether the probability model is realistic. You'll have to make a decision about how you want to model the behavior of two detectors. Are you dealing with a detector that is an actual physical device or are you dealing with some "concept" that appeared in an Operations Research paper? If you are only dealing with conceptual detector that is defined by an equation in an OR paper then you may be dealing with an concept that is absurd when it is carefully examined vis-a-vis the phenomena it supposedly represents. If you read different papers, you can't assume they are all dealing with the same concept in a consistent manner.

 

[chiro]

The probabiliy model does not address the issue of whether two detectors of the same target have independent probabilities of detecting it or not.

That is exactly what I was wondering.

For the OP it would really help to post all your variables with a one or two sample preview of the data you have, so this can be really clarified.

 

[Stephen Tashi]

I now notice that jingchyi87's original post says "dependent" bomb detectors. So the question will be to specifiy exactly how they are dependent.