5. Assume you are shooting at a target centred at the origin (0,0), but bullets land at a random point with horizontal (X) and vertical (Y) coordinates following in- dependent standard Normal (0,1) distributions. Define the distance from the tar- get to be D = √X² +Y2. For independent standard Normals, we already know that X²~ x²(1) = Gamma(1/2, 1/2) and X² + Y²~ x² (2) = Gamma(1, 1/2). (a) Find the probability that a bullet lands in the top right quadrant. (Hint: You don't need to calculate any integrals.) Y (b) Find the conditional probability that a bullet lands more than 1 unit away from the centre (D > 1), given Y = X, i.e. the bullet lands on the identity line; express the probability as a function of the standard Normal CDF Þ(.). (c) Find the CDF of X² + Y2 in closed form. (d) Find the probability that a bullet lands more than 1 unit away from the centre, P(D> 1). (e) Find the CDF of D based on the CDF of X² +Y². (f) Use your answer in the previous part to find the PDF of D. X
5. Assume you are shooting at a target centred at the origin (0,0), but bullets land at a random point with horizontal (X) and vertical (Y) coordinates following in- dependent standard Normal (0,1) distributions. Define the distance from the tar- get to be D = √X² +Y2. For independent standard Normals, we already know that X²~ x²(1) = Gamma(1/2, 1/2) and X² + Y²~ x² (2) = Gamma(1, 1/2). (a) Find the probability that a bullet lands in the top right quadrant. (Hint: You don't need to calculate any integrals.) Y (b) Find the conditional probability that a bullet lands more than 1 unit away from the centre (D > 1), given Y = X, i.e. the bullet lands on the identity line; express the probability as a function of the standard Normal CDF Þ(.). (c) Find the CDF of X² + Y2 in closed form. (d) Find the probability that a bullet lands more than 1 unit away from the centre, P(D> 1). (e) Find the CDF of D based on the CDF of X² +Y². (f) Use your answer in the previous part to find the PDF of D. X
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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Transcribed Image Text:5. Assume you are shooting at a target centred at the origin (0,0), but bullets land
at a random point with horizontal (X) and vertical (Y) coordinates following in-
dependent standard Normal (0,1) distributions. Define the distance from the tar-
get to be D = VX²+Y². For independent standard Normals, we already know
that X? - x²(1) = Gamma(1/2,1/2) and X² + Y² ~ x²(2) = Gamma(1, 1/2).
Y
/D
%3D
(a) Find the probability that a bullet lands in the top right quadrant.
(Hint: You don't need to calculate any integrals.)
(b) Find the conditional probability that a bullet lands more than 1 unit away from
the centre (D > 1), given Y = X, i.e. the bullet lands on the identity line; express the
probability as a function of the standard Normal CDF ¤(·).
(c) Find the CDF of X² + Y² in closed form.
(d) Find the probability that a bullet lands more than 1 unit away from the centre, P(D> 1).
(e) Find the CDF of D based on the CDF of X? + Y².
(f) Use your answer in the previous part to find the PDF of D.
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