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(b) Consider three (non-negative) random variables, X, Y, and Z. Let f(x, y, z) = kxyz² inside the following region E, and 0 elsewhere. The region E is defined as the region under the plane z = 4 - x - 2y and in the first octant (r ≥ 0, y ≥ 0, and z ≥ 0). What value must k take such that f(x, y, z) satisfies all of the conditions to be a probability density function? (c) Compute the probability P(Y > X). (d) Find the expected values of X, Y, and Z, denoting them as X, Ý, Ž respectively.

(b) Consider three (non-negative) random variables, X, Y, and Z. Let f(x, y, z) = kxyz² inside the following region E, and 0 elsewhere. The region E is defined as the region under the plane z = 4 - x - 2y and in the first octant (r ≥ 0, y ≥ 0, and z ≥ 0). What value must k take such that f(x, y, z) satisfies all of the conditions to be a probability density function? (c) Compute the probability P(Y > X). (d) Find the expected values of X, Y, and Z, denoting them as X, Ý, Ž respectively.

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter8: Further Techniques And Applications Of Integration
Section8.3: Volume And Average Value
Problem 11E
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Do parts b, c, d

(a) Give a qualitative interpretation of SSS fdV for each of the following descriptions of f. (i) f(x, y, z) = 1 for all points in a 3-dimensional region E. Hint: Recall the meaning of the double integral of f(x, y) = 1 (ii) f(x, y, z) is the mass density at point (x, y, z) of a solid object occupying a region E. (iii) f(x, y, z) is the joint probability density function for three random variables. (b) Consider three (non-negative) random variables, X, Y, and Z. Let f(x, y, z) = kxyz² inside the following region E, and 0 elsewhere. The region E is defined as the region under the plane z = 4-x-2y and in the first octant (r≥ 0, y ≥ 0, and z≥ 0). What value must k take such that f(x, y, z) satisfies all of the conditions to be a probability density function? (c) Compute the probability P(Y> X). (d) Find the expected values of X, Y, and Z, denoting them as X, Y, Z respectively.
Transcribed Image Text:(a) Give a qualitative interpretation of SSS fdV for each of the following descriptions of f. (i) f(x, y, z) = 1 for all points in a 3-dimensional region E. Hint: Recall the meaning of the double integral of f(x, y) = 1 (ii) f(x, y, z) is the mass density at point (x, y, z) of a solid object occupying a region E. (iii) f(x, y, z) is the joint probability density function for three random variables. (b) Consider three (non-negative) random variables, X, Y, and Z. Let f(x, y, z) = kxyz² inside the following region E, and 0 elsewhere. The region E is defined as the region under the plane z = 4-x-2y and in the first octant (r≥ 0, y ≥ 0, and z≥ 0). What value must k take such that f(x, y, z) satisfies all of the conditions to be a probability density function? (c) Compute the probability P(Y> X). (d) Find the expected values of X, Y, and Z, denoting them as X, Y, Z respectively.
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