Question

Given that \(f(x ; \theta)=\exp [\theta K(x)+S(x)+q(\theta)], a

Short answer

The moment-generating function \(M(t)\) of \(Y=K(X)\) is \(M(t)=\exp [q(\theta+t)-q(\theta)], \gamma<\theta+t<\delta\), as can be derived by using the definition of the moment-generating function, and recognizing the resulting integral as a normalizing constant for the given exponential function.

Step-by-step solution

Apply the definition of the moment-generating function

Start by noting the definition of the moment-generating function \(M(t)\) of a random variable \(Y\). It is defined as \(M(t) = E(e^{tY})\). So in this case, \(M(t) = E(e^{tK(X)}) = \int_{a}^{b} e^{tK(x)} f(x ; \theta) dx\).

Substitute the given function into the integral

With the given function \(f(x ; \theta)=\exp [\theta K(x)+S(x)+q(\theta)]\), the integral becomes \(M(t) = \int_{a}^{b} e^{tK(x)} \exp [\theta K(x) + S(x) + q(\theta)] dx = \int_{a}^{b} \exp [(t+\theta) K(x) + S(x) + q(\theta)] dx\).

Recognize the result as a normalizing constant

The result of the integral is similar in form to the given exponential function. It's the normalizing constant which ensures that \(f(x ; \theta)\) is a proper probability density function, that is, it integrates to 1 over the interval (a, b). Therefore, the result can be expressed as an exponential of the function \(q\), but with a different argument. So, \(M(t) = \exp[q(\theta+t) - q(\theta)].\) This result holds for \(\gamma < \theta+t < \delta\), which is the original range shifted by t.

Key concepts

Exponential Class

The exponential class of distributions is a versatile and widely utilized family of probability distributions in statistics. These distributions have a unique property: they can be expressed in a specific exponential form. In general, a function belongs to the exponential class if it can be represented as:\[f(x ; \theta) = \exp[\theta K(x) + S(x) + q(\theta)]\]where:

• \(\theta\) is the parameter associated with the distribution.
• \(K(x)\) is a function related to the data \(x\).
• \(S(x)\) is another function, which is part of the structure of \(f(x; \theta)\).
• \(q(\theta)\) is a function of the parameter \(\theta\). This term often plays a pivotal role in ensuring that the distribution is properly normalized.

This form is important because it helps in deriving various properties of statistical measures, such as the moment-generating function (MGF). The exponential class is not only mathematically convenient as seen in its functionality to facilitate computations like finding MGFs but also rich enough to include various known distributions, such as normal and exponential distributions.

Probability Density Function

A Probability Density Function, often abbreviated as PDF, is a function that describes the likelihood of a random variable to take on a particular value. For continuous random variables, the PDF provides a probability model for a range of values rather than a particular point.In the context of the exponential class, the PDF is given by the function \(f(x; \theta) = \exp[\theta K(x) + S(x) + q(\theta)]\), as outlined before. The core properties of a valid probability density function are:

• It must be non-negative for all values of \(x\).
• The total integral of the PDF over all possible values must be equal to 1. This ensures the entire probability space is covered.

This requirement is fulfilled through normalization, which involves the normalizing constant. Essentially, the normalizing constant scales the function so that its integral over the interval \([a, b]\) equals 1. Thus, the PDF not only describes the distribution of \(X\) effectively but also helps in deriving other statistical measures straightforwardly.

Normalizing Constant

The normalizing constant is a crucial part of any probability density function. Its role is to ensure that the integral of the density function over its entire range is equal to 1, satisfying the basic property of probabilities.In our given function, expressed in the exponential class form as \(f(x ; \theta) = \exp[\theta K(x) + S(x) + q(\theta)]\), the term \(q(\theta)\) plays the role of the normalizing constant. By rearranging the integral of the PDF such that its value becomes 1 over the interval \([a, b]\), the function becomes a valid PDF:\[\int_{a}^{b} \exp[\theta K(x) + S(x) + q(\theta)] \, dx = 1\]Interestingly, this has a direct implication when derived into the moment-generating function \(M(t) = \exp[q(\theta + t) - q(\theta)]\). Modifications in \(q(\theta)\) reflect changes in the integration interval, capturing the shifts and transformations in distribution parameter \(\theta\). Having a well-defined normalizing constant allows us to use MGFs and other statistical tools accurately while ensuring all probabilistic properties are maintained.