Question

Exercise 25. Show that the exponential distribution for time to failure has the orgetfulness property: (6.5)

Answer 1

P( T $\ge$ t₀ + t₁ | T $\ge$ t₀ )

= $\frac{P( T \ge t_0 + t_1 \cap T \ge t_0 ) }{P(T \ge t_0)}$

Now, the cdf of exponential distriution is,

P[T $\ge$ t] = $e^{-\lambda t}$ , t > 0

So, $\frac{P( T \ge t_0 + t_1 \cap T \ge t_0 ) }{P(T \ge t_0)}$

= $\frac{P( T \ge t_0 + t_1 ) }{P(T \ge t_0)}$

= $\frac{ e^{-\lambda (t_1+t_0)} }{e^{-\lambda t_0}}$

= $e^{-\lambda t_1}$

= P(T $\ge$ t₁) [Proved]