# Let X and Y be jointly continuous with joint probability ## Problem 32E Chapter 2.6

Statistics for Engineers and Scientists | 4th Edition

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Problem 32E

Let X and Y be jointly continuous with joint probability density function f(x, y) and marginal densities fX(x) and fY(y). Suppose that f(x, y) = g(x)h(y) where g(x) is a function of x alone, h(y) is a function of y alone, and both g(x) and h(y) are nonnegative.

a. Show that there exists a positive constant c such that fX(x) = cg(x) and fY(y) = (1/c)h(y).

b. Use part (a) to show that X and Y are independent.

Step-by-Step Solution:

Step 1 of 3</p>

a) Given probability density function  is a function of X alone is a function of Y alone = c When you differentiating the joint pdf  with respect to y

We get function of ‘X’ with some constant c

Step 2 of 3</p> = (1/c) When you differentiating the joint pdf  with respect to X

We get function of  ‘y’  with some constant 1/c

Step 3 of 3

##### ISBN: 9780073401331

The full step-by-step solution to problem: 32E from chapter: 2.6 was answered by Patricia, our top Statistics solution expert on 06/28/17, 11:15AM. This textbook survival guide was created for the textbook: Statistics for Engineers and Scientists , edition: 4th. This full solution covers the following key subjects: function, show, alone, let, densities. This expansive textbook survival guide covers 153 chapters, and 2440 solutions. The answer to “Let X and Y be jointly continuous with joint probability density function f(x, y) and marginal densities fX(x) and fY(y). Suppose that f(x, y) = g(x)h(y) where g(x) is a function of x alone, h(y) is a function of y alone, and both g(x) and h(y) are nonnegative.a. Show that there exists a positive constant c such that fX(x) = cg(x) and fY(y) = (1/c)h(y).________________b. Use part (a) to show that X and Y are independent.” is broken down into a number of easy to follow steps, and 76 words. Statistics for Engineers and Scientists was written by Patricia and is associated to the ISBN: 9780073401331. Since the solution to 32E from 2.6 chapter was answered, more than 231 students have viewed the full step-by-step answer.

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