In the study of statistics, a joint density function z=f(x,y) defined on a region in the x,y-plane is represented by asurface in space. The probability that asxsb and csysd is given by P(a≤x≤ b,csysd) below and is representedby the volume between the graph of f and the rectangular region given by asxsb and csysd. If f(x,y)= 1 is a jointdensity function, where 0≤x≤ 1 and 0 sys1, find P(x ≥ 1/5,y≥ 1/3)P(a≤x≤ b,csysd) =P(x≥ 1/5,y2 1/3) =(Simplify your answer.)d bff1(x,y)dxdyca

A function that describes the probability distribution of two or more random variables is called a…

Answered: In the study of statistics, a joint…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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We call p(x, y) the probability density function for attributes x and y if the probabilitythat an arbitrary member of a population has attribute x over the interval (a, b) as wellas attribute y over the interval (c, d), corresponds to the volume under the graph ofp(x, y) over the rectangle a ≤ x ≤ b, c ≤ y ≤ d, where the total volume under p(x, y)over the entire xy-plane is equal to 1 and p(x, y) ≥ 0 for all x and y. (a) Explain why the two conditions given at the end: “the total volumeunder p(x, y) over the entire xy-plane is equal to 1 and p(x, y) ≥ 0 for all x and y”are essential in the context of probabilities. Explain in simple terms (b) Express the probability (volume) referred to in the statement above as adouble integral. Intervals should correspond to the regionof integration and the order of integration are consistent and clearly indicated.(c) Use answer in part (b) withp(x, y) = (see image) to find 1. Verify that p(x, y) satisfies the definition (given above) of a…
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Let X, Y be random variables with joint density function f: a) Suppose Z = Y/X . Show that the density function of Z is (image 1) b) If X, Y are independent with distribution N(0, 1), then (image 2). Hint: Use part (a).
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