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I have a joint pdf $f(x_1,x_2,x_3) = e^{-x_1 -x_2 -x_3}I(x_1 \ge 0)I(x_2 \ge 0)I(x_3 \ge 0)$

and I want to calculate the probability $P(T_3 - T_2 \gt T_1)$ where $T_1 \lt T_2 \lt T_3$ are the order statistics.

so I calculate the joint pdf $g(t_1,t_2,t_3) = 6e^{-t_1 -t_2 -t_3}I(0 \le t_1 \le t_2 \le t_3)$

my integral is

$$\int_0^\infty\int_0^{t_3} \int_{0}^{t_3 - t_2} 6e^{-t_1 -t_2 -t_3}dt_1 dt_2 dt_3$$

would this be correct?

tim.mof
  • 59

1 Answers1

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Yes, that is correct (notice that they are in fact independent exponential random variables with rate 1).

QQQ
  • 751