The lifetime of a device has gamma distribution. Find the reliability of the device:

The hazard function increasing in time for

:

Find the reliability of two such devices in series:

Find the reliability of two such devices in parallel:

Compare the reliability of both systems for

and

:

A device has three lifetime stages: A, B, and C. The time spent in each phase follows an exponential distribution with a mean time of 10 hours; after phase C, a failure occurs. Find the distribution of the time to failure of this device:

Find the mean time to failure:

Find the probability that such a device would be operational for at least 40 hours:

Simulate time to failure for 30 independent devices:

In the morning rush hour, customers enter a coffee shop at a rate of 12 customers every 10 minutes. The time between customer arrivals follows an exponential distribution and the time between

arrivals follows a

GammaDistribution distribution. Find the probability of at least 40 customers arriving in 45 minutes:

Find the average waiting time until the 40

customer arrives:

Find the probability that the time until the 40

customer arrives is at least 1 hour:

Simulate the waiting time until the 40

customer arrives during rush hour over 30 days:

Mixtures of gamma distributions can be used to model multimodal data:

Histogram of waiting times for eruptions of the Old Faithful geyser exhibits two modes:

Fit a

MixtureDistribution to the data:

Compare the histogram to the PDF of estimated distribution:

Find the probability that the waiting time is over 80 minutes:

Find average waiting time:

Find most common waiting times:

Simulate waiting times for the next 60 eruptions:

LogNormalDistribution data can be modeled by a gamma distribution:

Compare the histogram to the PDF of estimated distribution:

Comparing log-likelihoods with estimation by lognormal distribution:

Stacy distribution is a special case of generalized

GammaDistribution: