Question

Any Stats people?
The time (in years) after reaching age 60 that it takes an individual to retire is approximately exponentially distributed with a mean of about 6 years. Suppose we randomly pick one retired individual. We are interested in the time after age 60 to retirement.
In a room of 1000 people over age 78, how many do you expect will NOT have retired yet?

Answer 1

For this problem, what is the exponential distribution?

Answer 2

1/6?

Answer 3

Or (1/6)(e^(-(1/6)x))

Answer 4

No, in general the exponential distribution is of the form
\[ P(x) = \lambda e^{-\lambda x } \]
What is lambda here ...

... right.

Now, in the equation you just wrote down, how do you interpret x?

Answer 5

The random variable X is time, in years, after age 60, that it takes an individual to retire

Answer 6

Right, so what is the probability that a person who is 78 will not have retired?

Answer 7

Um.... .95?

Answer 8

How'd you get that?

Answer 9

Integral (1/6e^((-1/6)x),x,18,infinity) and then 1 minus that...I'm not sure if this is right though.

Answer 10

So what does the probability distribution tell us? In other words, interpret P(x) in English.

Answer 11

P(x) is the probability that a person over the age of 78 would have retired.

Answer 12

No, P(x) is the distribution of retirement ages of people over 60. Hence
\[ \int_{18}^\infty P(x) \ dx \] is the proportion of people who retire at age 78 or higher.

Answer 13

Ok..so what I've found is the proportion of people who haven't retired.

Answer 14

Therefore 1 - that number = the proportion who have retired already

Answer 15

right.

Answer 16

*bookmark

Answer 17

So how do I find the exact number of people who've actually not retired yet?

Answer 18

E[number not retired | age > 78] = 1000 P(not retired | age > 78)

Now, how do you calculate that probability?

Answer 19

Ok...I solved it. The answer is 50 people.