Question concerning the expected position of an object

[Ryuzaki]

Suppose there's an object within a sphere of radius [itex]5[/itex]-metres from a given point [itex]P=(x_0,y_0,z_0)[/itex]. The probabilities of the object being within [itex]0-1[/itex], [itex]1-2[/itex], [itex]2-3[/itex], [itex]3-4[/itex] and [itex]4-5[/itex] metres of [itex]P[/itex] are given to be respectively [itex]p_1,p_2,p_3,p_4[/itex] and [itex]p_5[/itex]. With this information, is it possible to find the expected position of the object,i.e, its expected coordinates?

 

[Stephen Tashi]

You would have to make a specific assumption about the probability distribution of the object within each of those "shells".
If you assume a distribution that is spherically symmetric about (0,0,0) in each shell, the expected coordinates of the object will be (0,0,0).

 

[Ryuzaki]

What kind of an assumption do I need? Could you give an example? Also, if P = (0,0,0), how do you get the expected coordinates of the object to be (0,0,0)? Doesn't it depend on the values of the probabilities of the object being within each shell?

 

[Stephen Tashi]

What kind of an assumption do I need? Could you give an example?

Compute the volume v_i of each shell i = 1,2,3,4,5 and set the probability density function for the object within that shell to be p_i/v_i.

Also, if P = (0,0,0), how do you get the expected coordinates of the object to be (0,0,0)?

The expected value is (0,0,0) if the probability distributions are spherically symmetric. Think about a probability distribution on a line. If it is symmetric about x = 0 then the mean value of the distribution must be x = 0.

 

[CellCoree]

I would approach this question by first understanding the concept of expected position. Expected position is a statistical measure that represents the average location of an object based on its probability distribution. In this case, the probabilities of the object being within specific distances from point P provide us with a probability distribution.

Based on the given information, it is possible to find the expected position of the object. We can use the probabilities and the distances to calculate the weighted average of the coordinates. This can be done by multiplying each probability with its corresponding distance and then summing up the results. This will give us the expected coordinates of the object relative to point P.

However, it is important to note that this calculation will only give us an estimate of the object's expected position. The actual position of the object may vary due to factors such as measurement errors or external forces acting on the object. Therefore, it is important to consider the uncertainties associated with the expected position.

In conclusion, with the given information, it is possible to calculate the expected position of the object relative to point P. This can provide us with a useful estimate of the object's location, but it is important to acknowledge the uncertainties and limitations of this calculation. Further analysis and experimentation may be needed to refine the expected position and improve its accuracy.