For continuous random variables X and Y with joint probability density function f(x,y)={xe−(x+xy)x&gt;0 and y&gt;00otherwise Are X and Y independent? Explain.

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Brennan Flores

Answered question

2021-05-31

For continuous random variables X and Y with joint probability density function

f (x, y) = {\begin{cases} x e^{- (x + x y)} & x > 0 a n d y > 0 \\ 0 & o t h e r w i s e \end{cases}

Are X and Y independent? Explain.

Answer & Explanation

Latisha Oneil

Skilled2021-06-01Added 100 answers

The probability, $P (X > 1 a n d Y > 1)$ 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
$f (x, y) = x e^{- (x + x y)}$
$f_{X} (x) = e^{- x}$
$f_{Y} (y) = \frac{1}{(y + 1)^{2}}$
$f_{X} (x) \cdot f_{Y} (y) = \frac{e^{- x}}{(y + 1)^{2}}$
$\neq x e^{- (x + x y)}$
$\Rightarrow f (x, y) \neq f_{X} (x) \cdot f_{Y} (y)$
Thus, X and Y are not independent.

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