Question

Suppose X,Y,Z have joint pdf given by f_{X,Y,Z}(x, y, z) = k xyz if 0 <= x <= 1, 0 <= y <= 1, 0 <= z <= 1 ) and f_{X,Y,Z} = 0, otherwise. (a) Find k so that f_{X,Y,Z} is a genuine probability density function. (b) Are X,Y,Z independent? (c) Find P(X <= 1/2, Y <= 1/3, Z <= 1/4). (d) Find the marginal pdf f_{X,Y}(x, y). (e) Find the marginal pdf f_X(x).

          Suppose X,Y,Z have joint pdf given by

f_{X,Y,Z}(x, y, z) = k xyz if 0 <= x <= 1, 0 <= y <= 1, 0 <= z <= 1 ) and f_{X,Y,Z} = 0, otherwise.

(a) Find k so that f_{X,Y,Z} is a genuine probability density function.
(b) Are X,Y,Z independent?
(c) Find P(X <= 1/2, Y <= 1/3, Z <= 1/4).
(d) Find the marginal pdf f_{X,Y}(x, y).
(e) Find the marginal pdf f_X(x).

Suppose X,Y,Z have joint pdf given by

fX,Y,Z(x, y, z) = k xyz if 0 <= x <= 1, 0 <= y <= 1, 0 <= z <= 1 ) and fX,Y,Z = 0, otherwise.

(a) Find k so that fX,Y,Z is a genuine probability density function.
(b) Are X,Y,Z independent?
(c) Find P(X <= 1/2, Y <= 1/3, Z <= 1/4).
(d) Find the marginal pdf fX,Y(x, y).
(e) Find the marginal pdf fX(x).

Added by Derek R.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

Instant Answer

Solved by Expert Ameer Said

03/06/2022

Step 1

Step 1: To find the value of k such that f(x, z) is a genuine probability density function, we need to integrate the joint pdf f(x, y, z) over the entire space and set it equal to 1. Show more…

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Suppose X, Y, Z have a joint pdf given by f(x, y, z) = kxyz if 0 <= x <= 1, 0 <= y <= 1, 0 <= z <= 1 and f(x, y, z) = 0 otherwise. (a) Find k so that f(x, y, z) is a genuine probability density function. (b) Are X, Y, Z independent? (c) Find P(X <= 1/2, Y <= 1/3, Z <= 1/4). (d) Find the marginal pdf f(x, y). (e) Find the marginal pdf f(x).