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The exponential distribution is a probability distribution often used to model time until an event occurs, such as failure of a mechanical device or arrival of a phone call. In this context, the exponential distribution helps us understand how the parameter \( \lambda \), which is the average rate of defects per yard, behaves as a random variable. *Key characteristics of the exponential distribution:*

• The probability density function (PDF) is given by \( f(\lambda) = e^{-\lambda} \) for \( \lambda \geq 0 \).
• The exponential distribution is memoryless, meaning the probability of an event occurring in the future is independent of the past.
• It has one parameter: \( \lambda \), which represents the rate of occurrence.

Understanding the exponential distribution in this scenario helps us integrate over all possible rates \( \lambda \) to find the unconditional probability mass function of \( Y \), which counts the number of defects per yard.

The probability mass function (PMF) is a function that gives the probability that a discrete random variable is exactly equal to some value. For the Poisson distribution, which models the number of defects per yard \( Y \), the PMF can be expressed with a specific \( \lambda \) as follows:\[ P(Y = y | \lambda) = \frac{\lambda^y e^{-\lambda}}{y!} \]*Poisson distribution characteristics:*

• It describes the number of events occurring within a fixed interval of time or space.
• The events must happen independently of the time since the last event.
• Parameter \( \lambda \) is the rate at which events occur.

In our problem, since \( \lambda \) itself is random and follows an exponential distribution, the PMF for \( Y \) must account for this randomness through integration, eventually resulting in an unconditional or marginal PMF. This approach enables us to find an overall probability for any defect count \( y \), regardless of \( \lambda \)'s variability.

The Gamma function is an extension of the factorial function to non-integer values, and it is crucial in the integration process for non-integer cases. It is denoted by \( \Gamma(n) \), with the property that for positive integers \( n \), \( \Gamma(n) = (n-1)! \).*Properties and formulae:*

• The integral defining the Gamma function is \( \Gamma(a) = \int_0^{\infty} x^{a-1} e^{-x} \, dx \).
• The relation \( \Gamma(n) = (n-1)! \) holds true for natural numbers \( n \).

In the given problem, simplifying the integration of the density and PMF used the Gamma function because the integral \( \int_0^{\infty} \lambda^y e^{-2\lambda} \, d\lambda \) matches the form of a Gamma function when solving for the unconditional PMF of \( Y \).Utilizing the Gamma function effectively simplifies this computation, leveraging its properties to evaluate the otherwise complex integral.