Question

Alvin throws darts at a circular target of radius r and is equally likely to hit any point in the target. Let X be the distance of Alvin's hit from the center. (a) Find the PDF, the mean. and the variance of X. (b) The target has an inner circle of radius t. If X ? t, Alvin gets a score of S = 1/ X. Otherwise his score is S = O. Find the CDF of S. Is S a continuous random variable?

Answer 1

answer:

Since we have an X in the problem, let our coordinates be y and z.

We have uniform distribution over the circle. The area of a circle is ?r², so f(y,z) = 1/?r²

Then, if we convert to polar coordinates (as we are already using r, we will use R here),

f(R,?) = Rf(y,z) = R*1/?r² = R/?r² . Then, if we integrate over ? from 0 to 2?, f(R) = 2?R/?r² =

2R/r²

The mean is 2/3r - 0 = 2/3r

var(x) = E(x²) - (E(x))²

E(x²) = 1/2r² - 0 = 1/2r²

Then, var(X) = 1/2r² -  (2/3r)² = 1/18r²

b) I see a question mark in the if X?t. I am guessing that this is less than.

Then, S = 0 for X > t, and this has probability 1/?r²(?r² - ?t²) = 1 - t²/r²

F(R) =

Then, for X <= t, P(S <= s) = P(X >= 1/s) = 1 - P(X <= 1/s) = 1 - (1/s)²/r² = 1 - 1/s²r²

Thus, F(s) = 0, s<0

F(s) = 1 - t²/r² 0 <= s <= 1/t

F(s) = 1 - 1/s²r² s>= 1/t (Note that when s = 1/t, we get a value equal to F(0), which is what we would expect; also, the lim s->? F(s) = 1)

S is not continuous, as it jumps from 0 for S <0 to  1 - t²/r² at s = 0.