# Question concerning the expected position of an object

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1. Jan 25, 2015

### 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?

2. Jan 25, 2015

### 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).

3. Jan 30, 2015

### 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?

4. Jan 30, 2015

### Stephen Tashi

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.

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.