Question

"the life length x (in years) of a dvd player" is exponentially distributed with mean 5 years. what is the probability that a more than 5-year old dvd would still work for more than 3 years?

Answer 1

The exponential distribution is:

`\lambda e^(- \lambda x)`
where
`\lambda = (1)/(mean)=(1)/(5)`

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.

`P(A|B) = (P(A and B))/(P(B))`
where A is P(x>8) and B is P(x>5)
To find these probabilities, integrate over the distribution:

`P(x \ \textgreater \ N) = \int_N^(\infty) (1)/(5) e^(-x/5) dx = e^(-N/5)`

Sub into conditional probability formula:


`P(x \ \textgreater \ 8 | x\ \textgreater \ 5) = (P(x\ \textgreater \ 8))/(P(x\ \textgreater \ 5)) = (e^(-8/5))/(e^(-5/5)) = e^(-3/5) = 0.549`

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%