a. Explanation Given info: Let X and Y be jointly continuous with joint probability density function f (x, y) and marginal densities f X (x) and f Y (y) . Also, 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. Calculation: Let c = ∫ − ∞ ∞ h ( y ) d y The marginal density f X ( x ) of x is, f X ( x ) = ∫ − ∞ ∞ f ( x , y ) d y From given information, f ( x , y ) = g ( x ) h ( y ) Therefore f X ( x ) becomes, f X ( x ) = ∫ − ∞ ∞ g ( x ) h ( y ) d y = g ( x ) ∫ − ∞ ∞ h ( y ) d y = c g ( x ) Hence f X ( x ) = c g ( x ) b. To determine Show that X and Y are independent by using part (a).

4th Edition

ISBN: 9780073401331

Author: William Navidi Prof.

Publisher: McGraw-Hill Education

See similar textbooks

Question

Chapter 2.6, Problem 32E

To determine

Show that there exists a positive constant c such that

fX(x)=cg(x)fY(y)=1ch(y)

To determine

Show that X and Y are independent by using part (a).

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 2.6, Problem 31E

Chapter 2.6, Problem 33E

Students have asked these similar questions

Let X and Y be two continuous random variables with joint probability density function f(x,y) = k(x + y) for 0 < x < 1, 0 < y < 1 and 0 otherwise, where k is a constant. Find the marginal probability density functions of X and Y.

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.

Let X be a positive random variable having the probability density function fx(x) and 1 Y = be a decreasing function for 0

Chapter 2 Solutions

Statistics for Engineers and Scientists

Ch. 2.1 - The probability that a bearing fails during the...Ch. 2.1 - A die (six faces) has the number 1 painted on...Ch. 2.1 - A section of an exam contains four True-False...Ch. 2.1 - Three times each day, a quality engineer samples a...Ch. 2.1 - Four candidates are to be interviewed for a job....Ch. 2.1 - Refer to Exercise 5. Two candidates are randomly...Ch. 2.1 - In a survey of households with television sets,...Ch. 2.1 - An automobile insurance company divides customers...Ch. 2.1 - Among the cast aluminum parts manufactured on a...Ch. 2.1 - The article High Cumulative Risk of Lung Cancer...

Knowledge Booster

Show that there exists a positive constant c such that f X ( x ) = c g ( x ) f Y ( y ) = 1 c h ( y )

Solution Summary: The author explains that X and Y are jointly continuous with joint probability density function f (x, y) and marginal densities (g(x)h(y).

Show that there exists a positive constant c such that

fX(x)=cg(x)fY(y)=1ch(y)

Show that X and Y are independent by using part (a).

Want to see the full answer?

Chapter 2 Solutions

Show that there exists a positive constant c such that f X ( x ) = c g ( x ) f Y ( y ) = 1 c h ( y )

Solution Summary: The author explains that X and Y are jointly continuous with joint probability density function f (x, y) and marginal densities (g(x)h(y).

Show that there exists a positive constant c such that fX(x)=cg(x)fY(y)=1ch(y)

Show that X and Y are independent by using part (a).

Want to see the full answer?

Chapter 2 Solutions

Show that there exists a positive constant c such that

fX(x)=cg(x)fY(y)=1ch(y)