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).

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Author: Navidi

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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).

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Chapter 2.6, Problem 31E

Chapter 2.6, Problem 33E

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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)