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An ecologist wishes to select a point inside a circular sampling region according to a uniform distribution (in practice this could be done by first selecting a direction and then a distance from the center in that direction). Let X = the x coordinate of the point selected and Y = the y coordinate of the point selected. If the circle is centered at (0, 0) and has radius R, then the joint pdf of X and Y is given below. f(x, y) ={ R? x2 + y? s R? otherwise R (a) What is the probability that the selected point is within - of the center of the circular region? [Hint: Draw a picture of the region of positive density D. Because f(x, y) is constant on D, computing a probability reduces to computing an area.) (b) What is the probability that both X and Y differ from 0 by at most ? 0.3182 (c) What is the probability that both X and Y differ from O by at most 0.6364 (d) What is the marginal pdf of X? 2V R² – x? fAx) = What is the marginal pdf of Y? 2V R² – y² fAY) = Are X and Y independent? O No, X and Y are independent since f(x, y) = fx)f(v). O Yes, X and Y are independent since (x, y) - f(x)fAY). O Yes, X and Y are independent since f(x, y) - fy(x)fAY). O No, X and Y are not independent since f(x, y) = f(x)f(v).

An ecologist wishes to select a point inside a circular sampling region according to a uniform distribution (in practice this could be done by first selecting a direction and then a distance from the center in that direction). Let X = the x coordinate of the point selected and Y = the y coordinate of the point selected. If the circle is centered at (0, 0) and has radius R, then the joint pdf of X and Y is given below. f(x, y) ={ R? x2 + y? s R? otherwise R (a) What is the probability that the selected point is within - of the center of the circular region? [Hint: Draw a picture of the region of positive density D. Because f(x, y) is constant on D, computing a probability reduces to computing an area.) (b) What is the probability that both X and Y differ from 0 by at most ? 0.3182 (c) What is the probability that both X and Y differ from O by at most 0.6364 (d) What is the marginal pdf of X? 2V R² – x? fAx) = What is the marginal pdf of Y? 2V R² – y² fAY) = Are X and Y independent? O No, X and Y are independent since f(x, y) = fx)f(v). O Yes, X and Y are independent since (x, y) - f(x)fAY). O Yes, X and Y are independent since f(x, y) - fy(x)fAY). O No, X and Y are not independent since f(x, y) = f(x)f(v).

A First Course in Probability (10th Edition)
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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An ecologist wishes to select a point inside a circular sampling region according to a uniform distribution (in practice this could be done by first selecting a direction and then a distance from the center in that direction). Let X = the x coordinate of the point selected and Y = the y coordinate of the point selected. If the circle is centered at (0, 0) and has radius R, then the joint pdf of X and Y is given below. 1 f(x, y) = { TR2 x² + y2 < R? otherwise (a) What is the probability that the selected point is within of the center of the circular region? [Hint: Draw a picture of the region of positive density D. Because f(x, y) is constant on D, computing a probability reduces to computing an area.] (b) What is the probability that both X and Y differ from 0 by at most - 0.3182 R (c) What is the probability that both X and Y differ from 0 by at most 0.6364 (d) What is the marginal pdf of X? 2VR? -? TR? fx(x) = What is the marginal pdf of Y? 2VR?. fylY) = Are X and Y independent? O No, X and Y are independent since f(x, y) = f(x)f(y). O Yes, X and Y are independent since f(x, y) = fy(x)fy(y). O Yes, X and Y are independent since f(x, y) = fy(x)f(y). O No, X and Y are not independent since f(x, y) + fy(x)fy(y).
Transcribed Image Text:An ecologist wishes to select a point inside a circular sampling region according to a uniform distribution (in practice this could be done by first selecting a direction and then a distance from the center in that direction). Let X = the x coordinate of the point selected and Y = the y coordinate of the point selected. If the circle is centered at (0, 0) and has radius R, then the joint pdf of X and Y is given below. 1 f(x, y) = { TR2 x² + y2 < R? otherwise (a) What is the probability that the selected point is within of the center of the circular region? [Hint: Draw a picture of the region of positive density D. Because f(x, y) is constant on D, computing a probability reduces to computing an area.] (b) What is the probability that both X and Y differ from 0 by at most - 0.3182 R (c) What is the probability that both X and Y differ from 0 by at most 0.6364 (d) What is the marginal pdf of X? 2VR? -? TR? fx(x) = What is the marginal pdf of Y? 2VR?. fylY) = Are X and Y independent? O No, X and Y are independent since f(x, y) = f(x)f(y). O Yes, X and Y are independent since f(x, y) = fy(x)fy(y). O Yes, X and Y are independent since f(x, y) = fy(x)f(y). O No, X and Y are not independent since f(x, y) + fy(x)fy(y).
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