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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Answered question

2021-10-19

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

comentezq

Skilled2021-10-20Added 106 answers

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) = 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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