A battery has a lifespan that is exponentially distributed with parameter

. Find the probability that a random battery has a lifespan of less than 2500 hours:

Compute directly using the CDF:

A relay has an exponentially distributed lifetime with a failure rate of

failures per year. In order to estimate warranty costs, estimate the number of relays out of 10,000 that will fail in the first six months of use. The failure rate is also known as the hazard rate:

Hence the lifetime distribution (in years) for relays is:

The probability of failure within the first six months:

The expected number of failures within the first six months:

A product has a time to failure that is exponentially distributed with parameter

. Find the product's reliability at 1, 2, and 3 years. Reliability is another name for

SurvivalFunction:

What is the product's failure rate at 1, 2, and 3 years? Failure rate is also known as hazard rate:

Suppose the lifetime of an appliance has an exponential distribution with average lifetime of 10 years. Find the appliance lifetime distribution:

Find the probability that a used appliance with

years of use will not fail in the next 5 years:

Using the memoryless property of

ExponentialDistribution:

Assume the waiting time a customer spends in a restaurant is exponentially distributed with an average wait time of 5 minutes. Find the probability that the customer will have to wait more than 10 minutes:

Find the probability that the customer has to wait an additional 10 minutes, given that he or she has already been waiting for at least 10 minutes (the past does not matter):

The data contains waiting times in days between serious (magnitude at least 7.5 or over 1000 fatalities) earthquakes worldwide, recorded from 12/16/1902 to 3/4/1977:

Compare the histogram of the data with the PDF of the estimated distribution:

Find the average number of days between major earthquakes:

Find the probability that two serious earthquakes occur within 100 days:

Simulate times between the next 30 serious earthquakes occurring worldwide:

The fit is almost as good as the one with

DagumDistribution:

Waiting times at a receiver for signals coming from four independent transmitters are exponentially distributed with parameters

,

,

, and

, respectively. Find the probability that the signal from the third transmitter arrives first to the receiver:

Find the distribution of the waiting time for any signal at the receiver:

Find the average waiting time for any signal at the receiver:

Simulate the waiting time between signals arriving at the receiver for

,

,

, and

:

A system is composed of 4 independent components, each with lifespan exponentially distributed with parameter

. Find the probability that no component fails before 500 hours:

Find the probability that exactly one component will fail in the first 1200 hours:

Directly use

CDF and

SurvivalFunction:

By using

BooleanCountingFunction you can also define the logical condition:

In an optical communication system, transmitted light generates current at the receiver. The number of electrons follows the parametric mixture of Poisson distribution and other distributions, depending on the type of light. If the source uses coherent laser light of intensity

, then the electron count distribution is Poisson:

If the source uses thermal illumination, then the Poisson parameter follows

ExponentialDistribution with parameter

and the electron count distribution is:

These two distributions are distinguishable and allow the type of source to be determined: