Creating cumulative distribution function of Random variable.

• Oct 8th 2010, 11:17 PM
hotdone
Creating cumulative distribution function of Random variable.
Electrons hit a circular plate with unit radius. Let X be the random variable representing
the distance of a particle strike from the centre of the plate. Assuming that a particle is
equally likely to strike anywhere on the plate,
(a) for 0 < r < 1 find P(X < r), and hence write down the full the cumulative distribution
function of X, FX;
(b) find P(r < X < s), where r < s;
(c) find the probability density function for X, fX.
(d) calculate the mean distance of a particle strike from the origin.

I have problem in crating CDF and finding mean distance.
• Oct 8th 2010, 11:55 PM
CaptainBlack
Quote:

Originally Posted by hotdone
Electrons hit a circular plate with unit radius. Let X be the random variable representing
the distance of a particle strike from the centre of the plate. Assuming that a particle is
equally likely to strike anywhere on the plate,
(a) for 0 < r < 1 find P(X < r), and hence write down the full the cumulative distribution
function of X, FX;
(b) find P(r < X < s), where r < s;
(c) find the probability density function for X, fX.
(d) calculate the mean distance of a particle strike from the origin.

I have problem in crating CDF and finding mean distance.

The probability that P(X<r), 0<=z<=1 is the ratio of the area of a circle of radius r to that of one of radius 1. This is close to just the definition of a uniform distribution on the unit circle

CB
• Oct 9th 2010, 12:53 AM
hotdone
1)F(X)=(pi*r^2)/(pi*1)=r^2
ok so,
2)p(s>X>r)=(s^2-r^2)/r^2
3)p.d.f =d(F(X))/dx=2X
4)????(Worried)
• Oct 9th 2010, 02:39 AM
CaptainBlack
Quote:

Originally Posted by hotdone
1)F(X)=(pi*r^2)/(pi*1)=r^2
ok so,
2)p(s>X>r)=(s^2-r^2)/r^2
3)p.d.f =d(F(X))/dx=2X
4)????(Worried)

$\displaystyle \bar{x}=\int_0^1 x p(x)\;dx$

CB