# how to find f(x) and how to deal with the questions?

2021-11-16
Jones figures that the total number of thousands of miles that an auto can be driven before it would need to be junked is an exponential random variable with mean 20. Smith has a used car that he claims has been driven only 10,000 miles.
If Jones purchases the car, what is the probability that she would get at least 20,000 additional miles out of it?
Repeat under the assumption that the lifetime mileage of the car is not exponentially distributed, but rather is (in thousands of
miles) uniformly distributed over (0, 40).

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Let A be exponential random variable that represents the number of thousands of miles that a used auto can be
driven, $$X \sim exp(\frac{1}{20})$$
So what we want to calculate is probability that the car will cross 30 thousand miles if we have that it has already crossed 10 thousand miles:

$$P(X<30|X>10)=P(X>20+10|X>10)=P(X>20)=e^{-\frac{1}{20}\cdot 20}=0.368$$

Now let X be uniformly distriduted, $$X \sim U(0,40)$$. Now we have conditional probability:

$$P(X>30|X>10)=\frac{P(X>30)}{P(X>10)}=\frac{1-P(X \le 30)}{1-P(X \le 10)}=\frac{1-30/40}{1-10/40}=\frac{1}{3}$$