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:

g(x) = x² + 2

step-by-step explanation:

it is given that the graphs are the same shape.

the only difference is that the red graph is 2 units higher than the blue graph.

that is to say that the red graph is simply the blue graph that has been moved vertically by 2 units in the positive y direction.

hence

g(x) = f(x) + 2

or

g(x) = x² + 2