CDF of Distance and Angle from Origin of N(0, 1) RVs Y and Z

[london]

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

(a) Derive the CDF F_D(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 F_T (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.

 

[Ray Vickson]

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

(a) Derive the CDF F_D(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 F_T (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.

Show your work.

 

[london]

Show your work.

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?

 

[haruspex]

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?

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

 

[london]

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

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?

 

[Ray Vickson]

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?

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.

 

[berkeman]

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

(a) Derive the CDF F_D(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 F_T (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.

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.