Jeremy Yu Gong

Professor Ildar Gabitov

Math 485

15-19 March 2014

Proofs and Theory for the Probability Density

Classic analysis

I. Symmetric

As we started at any point of the road, by assuming probability density function

would generally result a symmetric distribution for the searching process. Secondly,

the searching process started by the center would go through each opposite direction

with equal search distance, which, the above two identities would have allowed us of

purely solving either side for the optimal result, typically as we are aiming at solving

for the shortest value of the expectation for the cost function.

II. Approach

There are however, two methods to approach the solution for the expectation for the

cost function: one is by assuming certain position of the objective, solving for the

expectation value for each single position, summing up all individual values by the

end; for instance:

Here we define the probability density and its anti-derivative:

For each certain assumed position of the objective with respect to the stepper index n,

we shall have the individual expectation value such that:

Where L is the total search distance for the single process:

Consequently, substituting L for the sum of each single step, we shall have the actual

form of the summation as follows:

For a final sum of all possible positions of the objective, one shall rearrange the above

expression in a better form such that:

The above equation yet is not the eventual answer we are looking for, since one shall

as well assuming for all the individual positions for the objective; evaluating each

single value for the above expression with respect to different possible position for

the objective; summing the following up to find the eventual value for the final

expectation of the cost function, hopefully solving for the conditions for the

minimization.

The above evaluation may seemingly straight forward for the problem approaches,

yet one obvious disadvantage is that, it takes too much steps for summation

expression, which, in turn, one shall eventually arrive at solving for a summation

within another summation; nonetheless, there is not yet a better arrangements for a

single summation term for the expression for the cost function. Consequently, a better

approach is required for the further calculation.

A generally tricky but better method to approach the problem is to assume the

potential position of the objective is lying in between each steps with respect to their

probabilities of actual being in the position.

Therefore, the total distance for the searching process in condition of the objective is

lying in between each searching steps is listed as follows, respectively:

In a brief conclusion:

Now, to treat the probability of the car lying in between of n-1 and nth step is

relevantly regarding the probability of taking the total searching distance as long as

indicates, thus a whole new rearrangement should be introduced to the solution

approach:

Hopefully, by the early definition of the anti-derivative of the probability density

function, we are able to drive the integral of each term in their actual from in order to

have a better rearrangement of expression for the cost function:

In the meantime, we could break the terms separately, and pair them in the following

manner:

2

For each sum of pairs of positive and negative , the cancellation is thus

available for driving the eventual form of the expression in terms of:

III. Analysis

As the above expression indicates, there are two summations for infinitely many

terms; yet as the probability density function could be randomly chosen, there are not

as much that one can do to drive a more concluded from of it; however, one typical

characteristic of the eventual expression would have granted us the possibility of

solving for the minimizing condition with computational simulations, as one shall

obviously notice that the terms of the expression is described as the difference

between two values, in turn, the value probability function is less than value one in

nature, which allows the function itself to converge to a certain value for the infinity

sums.

IV. Visibility

In practice, a sight parameter is required to further realize the modeling. The sight

parameter itself, must have physical affections to the construction of the equation that

the searching process is to be done whenever the objective were to within the visible

range. To describe it, instead of introducing an actual length of the visible sight, we

decided to covered it with a probability density function; one obvious reason for such

set up is due to the uninformed terrain along the searching process, thus any

assumptions for actual functions wouldn’t work well since. We therefore, define the

probability density for the visible range, say, the sight function as follows:

The above sight density function represents the probability of the actual length along

the searching process, with respect to the position of the detector; one shall notice,

however, several consequents result from the new definition which would cause

extremely problematic conditions for the further analysis:

1. Overlapping

By the looks of the probability density function, a minimum sight is ensured by

the delta function in the first place; the vision, by chance, could be extremely

long, which is capable of “seeing through” the whole distance to have the object

spotted, thus ends the searching process. For instance, even by sitting at the origin

of the searching process, the vision would still be extremely long, and spot the

objective before any steps were to be taken. Certain characters of the sight

function relevantly is quite well consistent with the actual saturation in reality,

since we might expect to have the car spotted in the first place if the weather is

not that bad, and the road is straightly long…etc.

2. Probability density function in nature

As the sight density function is constructed in the first place, we wouldn’t except

to have any analytic solutions that fit in certain conditions as might be needed in

further analysis; one for instance, the condition for if the objective is spotted

along the searching process is however, required for the evaluation of the

expectation value for the total distance (the cost function), yet the probability

density structure of the sight function provides quite an ambiguity for the

solution.

A clear definition for the above condition is thus required, we hence introduce the

“stopping time”, such that:

Let K be the first time that the objective is spotted along the search road, such that:

, where: is the probability function (anti-derivative of the sight

function) of the sight, l is the assumed position for the objective.

The above definition offers us a blink of chance for approaching to the eventual

results. Assumingly, the objective lies solidly at the position l along the road,

therefore, the chance of having the objective found in Kth (the stopping time) step

would be a production of the probability of objective to be found in Kth step, and the

probabilities of the objective not to be found before Kth step, which is shown as

follows:

Where: is the assumed probability function for the objective to be found with

respect to the position of the detector along the road; therefore, presents the

individual probability for the objective to be found until the Kth step of the searching

process, respectively, the total distance the detector would’ve travelled is to be:

Yet, the above probability function doesn’t yield the whole probability of this

saturation, since the objective would only have a “chance” of randomly lying on l,

one have to count such chance of occurrence:

Where l is not necessarily lying merely near Kth step, due to the overlapping, so that

the above integral is assumed to be evaluated under condition of the objective lying in

between Nth and N+1the step away from the Kth step (as we have already discussed

before, the overlapping character of the sight function provides extremely hash

condition for the further analysis). Whereby, the total distance we expected with

respect to the probability density function and the sight density function is to be the

sum of all those above terms from N=1 to infinity, and K=1, to K is equals to infinity,

shown as follows:

The above equation may serve as an analytic result for our problem, yet is proven to

be impracticable for any simulation, as one must notice that by calculating the whole

function, the “stopping time” must have a clear expression, so that the simulation

could appreciate the process, however, as the probability density function for the sight

in nature, the evaluation of the stopping time is almost impossible to execute.

Therefore, instead of the above conclusion, our calculation follows merely two

simply logic process:

1. Check at each position if the objective is spotted, if so, then end the searching

process.

2. If the objective is not yet within the visible sight, then continue the searching

process until the condition 1 occurs.

It might be seemingly strange for having such tedious calculation yet defines only

two logic process, however, the above logic process execute the evaluation process

well in computational simulations. As one may find hundreds or thousands of results

from the simulation as chaotic and inconvincible, yet a million or a billion results

would prove a general structure of the actual result of the searching process, and our

previous analytical result has provided a strong milestone for the necessity of

computational simulations to be involved.

Jeremy Yu Gong

Professor Ildar Gabitov

Math 485

27-28 April 2014

Proofs and Theory for the Probability Density

Sight Parameter

In practice, a sight parameter is required to further realize the modeling. The sight

parameter must have the physical affection to the construction of the equation that the

searching process is to be done whenever the objective were to within the visible

range. To describe it, we thus define the length of the visible range where, assumingly

the car is lying at , the alternated searching process would be executed as

follows:

3. What if the car was spotted when the detector were to arrive at , then finish the

whole searching process, the whole distance it had travelled (the cost) is

4. What if the car was not spotted when the detector were to arrive at , then keep

going through the process until it actually past the car; as the car lies in between

the same region, the exact total distance it had travelled is thus

Where, the length of sight is defined as follows:

Additionally, we defined the step function as a geometric series as follows:

To approach the expectation value for the cost function, we may start by the condition

of the sight function. It is obvious that there always exist a solution for in between

, assuming is extremely close to , then there always is:

Therefore, we define a position , such that:

For any in between the region where: , there exists the

solution to the function: , such that could be written as a

function of , where, for any , ;otherwise,

.

Our evaluation thus separated in conditions between two regions:

1. Inside the visible range:

2. Outside the visible range:

In order to make an easy life for the calculation, the above equation from the

second condition could be rearranged as follows:

Consequently, for every , and along the domain of our problem, there is:

The above formula could be further rearranged in much obvious manner, physically:

One must have noticed that the first term of the new cost function actually represent

the former cost function in our previous analysis, where, without introducing the sight

function, for each assumed position that the objective is lying in between the region

of , the cost function is:

Meanwhile, by introducing the sight function, so that the search process could be

fully executed when the objective was spotted before was passed by has reduced the

each individual term for the cost function by amount of:

With respect to its probability of occurrence, that:

The above equation could be further simplified, recalling for the calculation method

for the previous analysis:

2

However, for the second part, there barely is an alternation:

Where, for each