Question concerning the expected position of an object

  • #1
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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?
 

Answers and Replies

  • #2
Stephen Tashi
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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).
 
  • #3
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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?
 
  • #4
Stephen Tashi
Science Advisor
7,578
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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.
 

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