Question

Let \(X\) be a random variable with pdf of a regular case of the exponential class. Show that \(E[K(X)]=-q^{\prime}(\theta) / p^{\prime}(\theta)\), provided these derivatives exist, by differentiating both members of the equality $$\int_{a}^{b} \exp [p(\theta) K(x)+S(x)+q(\theta)] d x=1$$ with respect to \(\theta\). By a second differentiation, find the variance of \(K(X)\).

Short answer

The expected value of \(K(X)\) has been found to be \(-q^{\prime}(\theta) / p^{\prime}(\theta)\) by differentiating both sides of the given equation once. By doing a second differentiation, we obtain the needed elements to calculate the variance of \(K(X)\). The variance can then be found by using the equation for variance Var[X] = E[X^2] - (E[X])^2.

Step-by-step solution

Differentiate both sides of the equation

To begin, differentiate both sides of the equation \(\int_{a}^{b} \exp [p(\theta) K(x)+S(x)+q(\theta)] d x=1\) with respect to \(\theta\). Then, by the properties of the exponential function, this becomes: \(\int_{a}^{b} \exp [p(\theta) K(x)+S(x)+q(\theta)] \times (p'(\theta) K(x) + q'(\theta)) dx = 0\).

Rearrange the equation

The above equation can be rearranged to isolate \(E[K(x)]\) on one side. Subtracting \(q'(\theta)\) from both sides and dividing throughout by \(p'(\theta)\) gives us: \(E[K(X)]=-q^{\prime}(\theta) / p^{\prime}(\theta)\).

Differentiate a second time

To find the variance of \(K(X)\), differentiate the equation a second time. Using the rules of differentiation, this results in: \(\frac{d^2}{d\theta^2}\int_{a}^{b} \exp [p(\theta) K(x)+S(x)+q(\theta)] dx = 0\). This gives us the second moment.

Calculate the variance

The variance of \(K(X)\) can be found by subtracting the square of the expectation from the second moment, Var[X] = E[X^2] - (E[X])^2. The variance will be the final result.

Key concepts

Exponential Class

In statistics, when we say a random variable belongs to the exponential class, we're talking about a special type of probability distribution. The exponential class isn't just about the exponential distribution; it's a family of distributions that include exponential, Gaussian, Poisson, and many more. These distributions share a particular mathematical form.
The general form involves three key functions:

• \( p(\theta) \)
• \( K(x) \)
• \( S(x) + q(\theta) \)

Each of these functions helps define the behavior of the distribution. The exponential class is significant because it simplifies many statistical problems, especially those involving estimation and hypothesis testing. This special form allows for mathematical manipulation to derive useful properties such as expectation and variance, which are integral to understanding the properties of the random variable involved.

Variance Calculation

Variance is a measure of how spread out the values of a random variable are. For any random variable, calculating the variance involves two important steps: computing the expectation of the square of the variable and subtracting the square of the expectation. In mathematical terms, it's expressed as: \[\text{Var}[X] = E[X^2] - (E[X])^2\]To calculate the variance of a function of a random variable, like \( K(X) \), we start with the expectation formula for the exponential class. First, we differentiate the integral equation with respect to the parameter \( \theta \) a second time. By focusing on the resulting second derivative, which represents the second moment, we can direct our calculations towards variance.
Understanding variance is crucial for many applications in statistics as it provides a measure of uncertainty or risk, crucial for decision-making in fields such as finance, engineering, and research.

Probability Density Function

The Probability Density Function, or pdf, is a fundamental concept in probability and statistics. For a continuous random variable like \( X \), the pdf describes the likelihood of \( X \) assuming a particular value within a given range. Graphically, it's the curve that shapes the probability distribution of \( X \). The area under the pdf curve between two points \( a \) and \( b \) represents the probability that \( X \) falls within that interval.
One key property of the pdf is that integrating it over its entire range equals 1, ensuring the total probability is maintained. In the context of the exponential class, the pdf is often expressed in a specific structure involving \( \exp [p(\theta) K(x) + S(x) + q(\theta)] \), encapsulating the behavior of \( X \) concisely. This form is used to manipulate expressions mathematically, allowing us to derive expectations, variances, and other statistical measures efficiently.