Finding a Cumulative Distribution Function and Density Function

Let l n denote the line x = n in the Cartesian plane.

l n = {(n,t) : t E R}

A laser is positioned at the point (0,0). Suppose that the laser points into the first quadrant uniformly. Let X be the distance between (0,0) and the point that the laser hit on the line l n.

a) Find an explicit formula for the cumulative distribution function Fx (t) of X.

b) Find an explicit formula for the density function fx(t) of X.

Re: Finding a Cumulative Distribution Function and Density Function

Quote:

Originally Posted by

**bjnovak** Let l_n denote the line x = n in the Cartesian plane. L_n = {(n,t) : t E R}. A laser is positioned at the point (0,0). Suppose that the laser points into the first quadrant uniformly. Let X be the distance between (0,0) and the point that the laser hit on the line l_n.

a) Find an explicit formula for the cumulative distribution function Fx (t) of X.

b) Find an explicit formula for the density function fx(t) of X.

After reading this several times, it seems that some crucial statements(questions?) are missing. The setup is clear.

$\displaystyle X=n\sec(t)$, where $\displaystyle t\in\left(0,\tfrac{\pi}{2}\right)$ so that $\displaystyle X\in(n,\infty)$.

But what are the events in the space?

Re: Finding a Cumulative Distribution Function and Density Function

I don't know what to tell you, that's exactly how the problem was given.

Right, I understand X = n sec(t) , I think it's the fact that you have a uniform distribution so you have 1/(b-a) as being the area.

Then you can use the fact that X = n sec(t) to find the upper limit on the integral (since the cdf would be the integral and the density function is its derivative).

Re: Finding a Cumulative Distribution Function and Density Function

Quote:

Originally Posted by

**bjnovak** I don't know what to tell you, that's exactly how the problem was given.

Right, I understand X = n sec(t) , I think it's the fact that you have a uniform distribution so you have 1/(b-a) as being the area.

Then you can use the fact that X = n sec(t) to find the upper limit on the integral (since the cdf would be the integral and the density function is its derivative).

Sorry, but in the answer I gave, I use t in a different way that you posted.

If the endpoint is (n,t) then $\displaystyle X=\sqrt{n^2+t^2}$.

But still that means $\displaystyle t\in(0,\infty)~\&~X\in(n,\infty)$.

If if $\displaystyle t$ is uniform, I do not see the events? Do you?