Is Z Uniformly Distributed Between 0 and 1?

[gary.ming]

Homework Statement

2 independent r.v. X and Y, both of them are uniformly distributed on the interval [-1, 1]. a random variable Z is constructed by drawing samples from X and Y and forming X^2 + Y^2, however disregarding any draws that give Z > 1. Show that Z is uniformly distributed on [0,1].

Homework Equations

My understanding is Z is drawing from its samples from a square with area 4. Proof that the radius of the unit circle inside the square is uniformly distributed.

The Attempt at a Solution

Any idea, please?

 

[mighty2000]

Thank you for your question. I am happy to help you understand why Z is uniformly distributed on [0,1].

First, let's review the properties of a uniform distribution. A random variable X is said to be uniformly distributed on the interval [a,b] if its probability density function is given by f(x) = 1/(b-a) for a ≤ x ≤ b and 0 otherwise. This means that every value in the interval has an equal chance of being selected.

Now, let's consider the construction of Z. We know that X and Y are uniformly distributed on [-1,1], and Z is formed by taking the sum of their squares. This means that Z can only take on values between 0 and 2, as the maximum value for X^2 and Y^2 is 1 each.

However, we are disregarding any draws that give Z > 1. This means that the only values of Z that can occur are between 0 and 1, as any values greater than 1 are not included in our sample.

Now, let's consider the distribution of Z. Since we are taking the sum of squares of two uniformly distributed variables, we can see that Z is essentially the squared distance from the origin (0,0) to a point (X,Y) in a unit square.

To visualize this, imagine the unit square with X and Y as the coordinates of a point inside the square. If we draw a circle with radius 1 centered at the origin, we can see that the points within this circle are the only ones that will give Z ≤ 1. This is because any points outside of the circle will have a distance greater than 1 from the origin, and thus will be disregarded.

Since the area of the unit circle is π, and the area of the unit square is 4, we can see that the probability of Z being within the unit circle is π/4, which is equivalent to the probability of Z being between 0 and 1. This means that Z is indeed uniformly distributed on [0,1].

I hope this explanation helps you understand why Z is uniformly distributed on [0,1]. If you have any further questions, please do not hesitate to ask.