So I have an assignment regarding probability and random variables which states the following:
The random variable Y is defined in the following statement:
- On a random way we select a point from a circle with a radius of $6$.
- The random variable $X$ is defined as the distance between the selected point and the center of the circle and $Y = 3 - X$.
- Using the following statements define the function of distribution and the density of the distribution for the random variable $Y$.
I assume since we pick the point at random and each point has a fair chance of getting picked the Y has a uniform distribution with the constant c being equaled to 1/36*PI.
However I have no idea how to implement the random variable X or the statement Y=3-X in the equation.
Can anyone help me?
It is true that the uniform random distribution of points inside the circle has a factor of $\dfrac{1}{36\pi}.$ That is, if someone has in mind a particular subset of points within the circle and that subset of points has area $A,$ then the probability that the randomly selected point will be in that particular subset is $\dfrac{A}{36\pi}.$
But you have been asked for a random distribution over numbers, and points are not numbers.
What is the minimum value of $X$? Since $X$ is the distance between the selected point and the center of the circle, if the selected point is the center of the circle then $X=0.$ You can't have a distance less than $0,$ so that's the minimum value of $X.$
What is the maximum value of $X$? Can $X = 36\pi$? If you think it can, find a point that could be selected to make $X = 36\pi.$ Can $X = 7$?
Find the set of points such that $X \leq 5.$ What is the area of that set? What is the probability that $X \leq 5$? Try this for some other numbers instead of $5.$ Do you see a pattern?