The exponential distribution is:
[tex]\lambda e^{- \lambda x}[/tex]
where [tex]\lambda = \frac{1}{mean}=\frac{1}{5}[/tex]
The probability we want is how likely will a dvd player last more than 8 years, given it has already lasted 5 years
To find this, you use conditional probability.
[tex]P(A|B) = \frac{P(A and B)}{P(B)} [/tex]
where A is P(x>8) and B is P(x>5)
To find these probabilities, integrate over the distribution:
[tex]P(x \ \textgreater \ N) = \int_N^{\infty} \frac{1}{5} e^{-x/5} dx = e^{-N/5} [/tex]
Sub into conditional probability formula:
[tex]P(x \ \textgreater \ 8 | x\ \textgreater \ 5) = \frac{P(x\ \textgreater \ 8)}{P(x\ \textgreater \ 5)} = \frac{e^{-8/5}}{e^{-5/5}} = e^{-3/5} = 0.549[/tex]
Final Answer: Given a dvd player is more than 5 years old, the probability that it will last another 3 more years is about 54.9%
