Random Coordinate in a Sphere

Quant Placement Test question asked in APT Portfolio

Calvin randomly choses a coordinate (x, y, z) on the unit sphere‌‌‌‌‍‌‌‍‌‌‍‌‍‍‌‍‍‌‌. What is the variance of the z-coordinate?

Try to solve this problem yourself before moving on to the solution below

.
.
.

.

.

.

---

Solution

This is a an interesting problem solved using the basic concepts.
Using the standard variance equation we get:

\(Var(z) = E[z^2] - E[z]^2\)

We know the equation of a the unit sphere is:

\(x^2 +y^2 +z^2 =1\)

Since the coordinates are uniformly and independently selected, we can use the property of symmetry on all axes and state that the z^2 will make the third part of this. Hence:

\(E[z^2] = \frac{1}{3}\)

To find E[z], we can imagine the unit sphere. The average of all coordinates will be in the exact centre at (0,0,0) (0,0,0), giving us our expected value of z.

\(Var(z) = \frac{1}{3} - 0^2 = \frac{1}{3}\)

.

.

😃 Over to you: Were you able to solve this?

---

👉 If you liked this post, don’t forget to leave a like ❤️. It helps more people discover this newsletter on Substack and tells us that you appreciate solving these weekly questions. The button is located towards the bottom of this email.

👉 If you love reading this newsletter, feel free to share it with friends!