In a game Alis and Daniel shoot arrows on a circular target with a
radius of R.Alis throws arrows such that their distance from the center have
Continuous uniform distribution (0,R) – o,R are the parameters for
uniform distribution.While Daniel throws arrows such that their distance from the center have
random Continuous uniform distribution (From the
target)
I was asked to calculate the expected value for the distance of the arrows for both participant.
For Alis it's R/2 but what about Daniel? I didn't understand what's given about him, what are the parameters for him…
I claimed: "For Alis it's R/2" Proof:
The expected value for random variable with uniform distribution (a,b) (with parameters a and b) is (a+b)/2
Best Answer
HINT: Presumably it means that the probability that Daniel’s arrow lands in a given region of the target is proportional to the area of the region. The area of an annulus of mean radius $r$ and width $dr$ is $dA=2\pi r\,dr$, and the area of the target is $\pi R^2$, so the probability that it lands in that annulus is
$$\frac{dA}{\pi R^2}=\frac{2\pi r\,dr}{\pi R^2}=\frac2{R^2}r\,dr\,.$$
Alternatively, the cumulative distribution function $F(r)$ for $0\le r\le R$ is given by
$$F(r)=\frac{\pi r^2}{\pi R^2}=\frac{r^2}{R^2}\,.$$
Use whichever approach you like to get the probability density function, and then use that to calculated the expected value from the definition of expected value.