Under average driving conditions, the life lengths of automobile tires of a certain brand are found to follow an exponential distribution, with a mean of 30,000 miles. find the probability that one of these tires, bought today, will last the following number of miles: a. over 30,000 milesb. over 30,000 miles, given that it already has gone 15,000 miles.

a)

b)

Step-by-step explanation:

Previous concepts

The exponential distribution is "the probability distribution of the time between events in a Poisson process (a process in which events occur continuously and independently at a constant average rate). It is a particular case of the gamma distribution". The probability density function is given by:

And 0 for other case. Let X the random variable that represent "life lengths of automobile tires of a certain brand" and we know that the distribution is given by:

The cumulative distribution function is given by:

Part a

We want to find this probability:

and for this case we can use the cumulative distribution function to find it like this:

Part b

For this case w want to find this probability

We have an important property on the exponential distribution called "Memoryless" property and says this:

On this case if we use this property we have this:

We can use the definition of the density function and find this probability:

2500

step-by-step explanation: