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.
(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.
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