Probability | Bivariate Random Distribution

Question

Answer 1

$P(x>6/x>0) = ?$

For x > 0, we have to consider, f(x) = $ce^{-x/3}$

 

$P(x>6 / x>0) = P(x>6 \cap x>0)/P(x>0)$

 

$P(x>0) = \int_{0}^{\infty}ce^{-x/3} = c \int_{0}^{\infty} e^{-x/3} = -3c {e^{-x/3}} |^{\infty } _{0} = 3c$

$P(x>6) = \int_{6}^{\infty}ce^{-x/3} = c \int_{6}^{\infty} e^{-x/3} = -3c {e^{-x/3}} |^{\infty } _{6} = 3ce^{-2}$

 

Now, 

$P(x>6 / x>0) = P(x>6 \cap x>0)/P(x>0)$ = $P(x>6)/P(x>0)$ = $(3ce^{-2})/ 3c  =e^{-2}$

 

Hence, option A is the correct answer.