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 Find P(X&gt;1 and Y&gt;1).

The probability, P (X &gt; 1 and Y &gt; 1) can be calculated using the joint probability density function. The calculation of P (X &gt; 1 and Y &gt; 1) is shown below: P(X&gt;1 and Y&gt;1)=P(X&gt;1,Y&gt;1) =∫1∞∫1∞f(x,y)dxdy =∫1∞∫1∞xe−(x+xy)dxdy =∫1∞xe−x(∫1∞e−xydy)dx =∫1∞xe−x([e−xy−x]1∞)dx =∫1∞xe−x(e−xx)dx =∫1∞e−2xdx =[e−2x−2]1∞ =12e−2 ≈0.0677. Thus, the value of P (X &gt; 1 and Y &gt; 1) is (e−2)2, or approximately 0.0677.

"For continuous random variables X and Y with..." solved and explained

Dolly Robinson

Answered question

2021-11-03

For continuous random variables X and Y with joint probability density function
$f (x, y) = {\begin{cases} x e^{- (x + x y)} & x > 0 a n d y > 0 \\ 0 & o t h e r w i s e \end{cases}$
Find P(X>1 and Y>1).

Answer & Explanation

wornoutwomanC

Skilled2021-11-04Added 81 answers

The probability, P (X > 1 and Y > 1) can be calculated using the joint probability density function.
The calculation of P (X > 1 and Y > 1) is shown below:
P(X>1 and Y>1)=P(X>1,Y>1)

= \int_{1}^{\infty} \int_{1}^{\infty} f (x, y) d x d y

= \int_{1}^{\infty} \int_{1}^{\infty} x e^{- (x + x y)} d x d y

= \int_{1}^{\infty} x e^{- x} (\int_{1}^{\infty} e^{- x y} d y) d x

= \int_{1}^{\infty} x e^{- x} ({[\frac{e^{- x y}}{- x}]}_{1}^{\infty}) d x

= \int_{1}^{\infty} x e^{- x} (\frac{e^{- x}}{x}) d x

= \int_{1}^{\infty} e^{- 2 x} d x

= {[\frac{e^{- 2 x}}{- 2}]}_{1}^{\infty}

= \frac{1}{2} e^{- 2}

\approx 0.0677

.
Thus, the value of P (X > 1 and Y > 1) is

\frac{(e^{- 2})}{2}

, or approximately 0.0677.

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