Question concerning the expected position of an object

[Ryuzaki]

Suppose there's an object within a sphere of radius $5$-metres from a given point $P=(x_0,y_0,z_0)$. The probabilities of the object being within $0-1$, $1-2$, $2-3$, $3-4$ and $4-5$ metres of $P$ are given to be respectively $p_1,p_2,p_3,p_4$ and $p_5$. 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.