Final answer:
The underlying distribution for the c-chart is the Poisson distribution, which is used to model the number of defects per unit in a fixed interval when events occur independently and at a known rate.
Step-by-step explanation:
The underlying distribution for the c-chart is the Poisson distribution. This is because a c-chart is used to monitor the number of defects per unit of output, and the Poisson distribution is suitable for modeling the number of events or occurrences in a fixed interval of time or space if these events happen with a known average rate and independently of the time since the last event.
The other options mentioned do not apply to the c-chart: the exponential distribution is used for the time between events in a Poisson process; the normal distribution is a continuous distribution used for data that can be thought of as the sum of many small, independent random variables; and the gamma distribution (γ) is a two-parameter family of continuous probability distributions, which is not a common underlying distribution for the c-chart.
There's an interesting relationship between the Poisson and the exponential distributions. The number of events in a Poisson process over a fixed amount of time is Poisson distributed, and the time between these events follows an exponential distribution. This property is crucial in scenarios like modeling the time between arrivals at a service point or the time between failures in a manufacturing process.