Question

For continuous random variables X and Y with joint probability density function f(x,y)=\begin{cases}xe^{-(x+xy)} & x>0\ and\ y>0\\0 & otherwise\end{cases} Are X and Y independent? Explain.

Random variables
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asked 2021-05-31
For continuous random variables X and Y with joint probability density function
\(f(x,y)=\begin{cases}xe^{-(x+xy)} & x>0\ and\ y>0\\0 & otherwise\end{cases}\)
Are X and Y independent? Explain.

Expert Answers (1)

2021-06-01

The probability, \(P (X > 1\ and\ Y > 1)\) can be calculated using the joint probability density function.
If two random variables, X and Y are independent, then the joint density function can be written as a product of the marginal density function, that is,
\(f(x,y)=f_{X}(x)\cdot f_{Y}(y)\)
Here
\(f(x,y)=xe^{-(x+xy)}\)
\(f_{X}(x)=e^{-x}\)
\(f_{Y}(y)=\frac{1}{(y+1)^{2}}\)
\(f_{X}(x)\cdot f_{Y}(y)=\frac{e^{-x}}{(y+1)^{2}}\)
\(\neq xe^{-(x+xy)}\)
\(\Rightarrow f(x,y)\neq f_{X}(x)\cdot f_{Y}(y)\)
Thus, X and Y are not independent.

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