Uniform Probability Density Function

[cse63146]

Homework Statement

Suppose that a point $$(X_1 , X_2 , X_3)$$ is chosen at random, that is, in accordance with the uniform probability function over the following set S:

$$S = {(x_1, x_2, x_3) : 0 \leq x_i \leq 1 for i =1,2,3}$$

Determine $$P[(X_1 - 1/2)^2 + (X_2 - 1/2)^2 + (X_3 - 1/2)^2) \leq 1/4]$$

Homework Equations

The Attempt at a Solution

Not sure how to set this up. Any suggestions/hints?

 

[lanedance]

Hi cse63149

the allowable variable space defines a cube... I'd try thinking about volume and how you could relate volume it to probability...

 

[cse63146]

Would it be a series of integrals or something?

Sorry, still not getting it.

 

[HallsofIvy]

Not a series of integrals- a single integral. The volume of the given cube is 1 so the probability that $(x- 1/2)^2+ (y- 1/2)^2+ (z- 1/2)^2\le 1/4$ is is the volume of that figure.

 

[cse63146]

so:

$$\int^{1/4}_0 (X_1 - 1/2)^2 dX_1 + \int^{1/4}_0 (X_2 - 1/2)^2 dX_2 + \int^{1/4}_0 (X_3 - 1/2)^2 dX_3$$

Is that correct?

 

[Mark44]

What geometric figure does this inequality describe?
$(x- 1/2)^2+ (y- 1/2)^2+ (z- 1/2)^2\le 1/4$
And what is the volume of this figure?

 

[cse63146]

The shape is a sphere, but wouldn't the triple I set up in my earlier post be correct, or did I make a mistake?

 

[Mark44]

No, they're not correct. All three produce the same value, the area under a parabola, and you're adding these identical areas to get, what?

Your probability represents a sphere of a certain size, within a cube-shaped region with volume 1. You don't need any calculus for this problem, just a bit of garden-variety geometry.

 

[cse63146]

Sorry again, but I'm still stuck. Would it be possible to get another hint?

 

[Mark44]

When you get it, you're going to slap yourself in the forehead and say "Doh!"

The probability you're trying to find in this uniform distribution is nothing more than the ratio of the volume of the ball to the volume of your 1 x 1 x 1 cube. (A ball is a solid object, all of whose surface points are equidistant from its center. A sphere is the skin of a ball.)

The ball is the set $(x- 1/2)^2+ (y- 1/2)^2+ (z- 1/2)^2\le 1/4$.
From this inequality, I can see by inspection the center of the ball and its radius, so I pretty much know everything there is to know about the ball.

 

[cse63146]

I got the ratio part, but what I'm trying to find is the volume of the ball, unless it's in fact 1/4 (which I doubut it is)

would the center of the ball be (1/2, 1/2, 1/2) and its radius is 1/8?

 

[lanedance]

getting close radius looks a bit off... the equation of a sphere at center (a,b,c) and radius r will look like

$$(x-a)^2 + (y-b)^2 + (y-c)^2 = r^2$$

to understand the consider the points defined setting y=b, z=c
then
(x-a)^2 = r^2

ie the distance for x to a is the radius as expected...

 

[cse63146]

since $$r^2 = 1/4 \rightarrow r = 1 / \sqrt{4} \rightarrow r = 1/2$$

 

[lanedance]

looks good

so think about how the sphere of centre (1/2,1/2,1/2) and radius 1/2 sits in the cube

first you should ask is it fully contained in the cube?

then what are the ratio of volumes?

as the variable are uniformly distributed, this ratio should give you yoru probability

 

[cse63146]

Might seem like a stupid question at this point, how would I find the volume of the sphere?

 

[Mark44]

For a sphere, V = 4/3 * pi * r^3

 

[cse63146]

For a sphere, V = 4/3 * pi * r^3

*bangs head repeteadly*

$$\frac{volume of sphere}{volume of cube} = \frac{ (4/3) \pi (1/2)^3}{1} ~ \frac{0.5236}{1} = 0.5236$$

 

[Mark44]

Like I said...

 

[cse63146]

Like I said...

yeah, sorry. I only realized it during class the sphere isn't fully contained inside the cube, and I didn't have access to the internet to fix my mistake

since the sphere is centred at (1/2,1/2,1/2), only half of it would be inside the cube it should be:

$$\frac{volume of sphere}{volume of cube} =\frac{\frac{volume of sphere}{2}}{volume of cube} = \frac{ ((4/3) \pi (1/2)^3)/2}{1} ~ \frac{0.2618}{1} = 0.2618$$

hopefully I didnt make another stupid mistake somewhere

 

[Mark44]

The ball is of radius 1/2 and its center is at (1/2, 1/2, 1/2). It is entirely inside (or just touching) the cube.

 

[cse63146]

in that case, I'm not sure where I made the mistake in:

$$\frac{volume of sphere}{volume of cube} =\frac{\frac{volume of sphere}{2}}{volume of cube} = \frac{ ((4/3) \pi (1/2)^3)/2}{1} ~ \frac{0.2618}{1} = 0.2618$$

 

[lanedance]

why did you change to dividing the sphere in half? It is fully contained in the cube (try drawing it)

 

[cse63146]

I was thinking of something else.

I thought I made some sort of mistake whem mark said "Like I said" but I think he's referring to the part of me banging my head (which I actually did)

 

[lanedance]

i know the feeling...

 

[cse63146]

Just to make sure, the answer of 0.5236 is the correct one, right?

 

[Mark44]

Looks good.

 

[cse63146]

Thank You both.