# Statistics (CDF)

1. Sep 19, 2014

### london

Y and Z are independent N(0, 1) random variables. Let X = |Z|. Consider the random point (X, Y).

(a) Derive the CDF FD(d) = P(D ≤ d) of the distance from the origin D = √(X2 + Y2). Sketch this CDF as a function of all real d.

(b) The ratio T = Y/X has Student’s t-distribution with 1 degree of freedom, also called the Cauchy distribution with CDF FT (t) = P(T ≤ t) = 1/2 + 1/π tan−1(t). Use this to determine the CDF FA(a) = P(A ≤ a) of the random angle A = tan−1(Y/X) between the line joining the origin and (X, Y) and the X-axis, for −π/2 < a < π/2 (points below the X-axis subtend a negative angle). Sketch this CDF as a function of all real a.

(c) Determine the probability P(Y > cX) for a constant c.

2. Sep 19, 2014

### Ray Vickson

3. Sep 19, 2014

### london

well i really have no idea where to start :/
But i was thinking maybe of using joint distributions but that leads to double integrals and i have not been taught double integrals so there must be another way about it right?

4. Sep 19, 2014

### haruspex

Yes, it involves a double integral, but that should not stop you at least writing out the expression for it.

5. Sep 19, 2014

### london

but what's the point of writing it out when its supposed to be derived so I can sketch it? My lecturer said double integrals are not assessable so clearly another method is used???

6. Sep 19, 2014

### Ray Vickson

You need to do some work here. If there ARE some tools/results you are allowed to use you need to tell us about them. We cannot possibly help if we have zero information.

I urge you to read Vela's 'pinned' thread "Guidelines for students and helpers', which explains what the expectations are when you post to this Forum. In particular, stating that you have no idea how to start is not acceptable.

7. Sep 19, 2014

### Staff: Mentor

Please check your PMs. Per the PF rules (see Site Info at the top of the page), you *must* show your efforts toward solving the problem before we can offer tutorial help.