Solution Found!
Let X and Y be jointly continuous with joint probability
Chapter 2, Problem 32E(choose chapter or problem)
Let and
be jointly continuous with joint probability density function \(f(x, y)\) and marginal densities \(f_{X}(x)\) and \(f_{Y}(y)\). Suppose that \(f(x, y)=g(x) h(y)\) where \(g(x)\) is a function of
alone, \(h(y)\) is a function of
alone, and both \(g(x)\) and \(h(y)\) are nonnegative.
a. Show that there exists a positive constant such that \(f_{X}(x)=c g(x)\) and \(f_{Y}(y)=(1 / c) h(y)\).
b. Use part (a) to show that X and Y are independent.
Equation Transcription:
Text Transcription:
f(x,y)
f_X(x)
f_Y(y)
f(x,y)=g(x)h(y)
g(x)
h(y)
g(x)
h(y)
f_X(x)=cg(x)
f_Y(y)=(1/c)h(y)
Questions & Answers
QUESTION:
Let and
be jointly continuous with joint probability density function \(f(x, y)\) and marginal densities \(f_{X}(x)\) and \(f_{Y}(y)\). Suppose that \(f(x, y)=g(x) h(y)\) where \(g(x)\) is a function of
alone, \(h(y)\) is a function of
alone, and both \(g(x)\) and \(h(y)\) are nonnegative.
a. Show that there exists a positive constant such that \(f_{X}(x)=c g(x)\) and \(f_{Y}(y)=(1 / c) h(y)\).
b. Use part (a) to show that X and Y are independent.
Equation Transcription:
Text Transcription:
f(x,y)
f_X(x)
f_Y(y)
f(x,y)=g(x)h(y)
g(x)
h(y)
g(x)
h(y)
f_X(x)=cg(x)
f_Y(y)=(1/c)h(y)
ANSWER:
Answer
Step 1 of 3
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