Question

Let \(\psi(t)=\log M(t)\), where \(M(t)\) is the mgf of a distribution. Prove that \(\psi^{\prime}(0)=\mu\) and \(\psi^{\prime \prime}(0)=\sigma^{2}\). The function \(\psi(t)\) is called the cumulant generating function.

Short answer

The first derivative of the cumulant generating function evaluated at zero gives the mean of the distribution. The second derivative evaluated at zero gives the variance of the distribution. Thus, the results \(\psi^{\prime}(0)=\mu\) and \(\psi^{\prime\prime}(0)=\sigma^{2}\) are proved.

Step-by-step solution

Calculate the derivatives of \(\psi(t)\)

The derivatives of \(\psi(t)\) are calculated using the basic rules of differentiation and the chain rule. The first derivative, \(\psi^{\prime}(t)\), is the derivative of \(\log M(t)\) which is \(\frac{M^{\prime}(t)}{M(t)}\). The second derivative, \(\psi^{\prime \prime}(t)\), is the derivative of the first derivative, which is \(\frac{M^{\prime\prime}(t)M(t) - [M^{\prime}(t)]^2}{[M(t)]^2}\).

Evaluate the derivatives at \(t=0\)

Plug in \(t=0\) into the derivatives. We know that the MGF evaluated at 0 (\(M(0)\)) is always 1. Thus, \(\psi^{\prime}(0) = M^{\prime}(0) = \mu\), and \(\psi^{\prime\prime}(0) = M^{\prime\prime}(0)- [M^{\prime}(0)]^2 = \sigma^{2}\).

Interpret results

The first derivative of the cumulant generating function evaluated at zero gives us the mean of the distribution, which is consistent with the first moment about the origin. Similarly, the second derivative evaluated at zero gives us the variance of the distribution, consistent with the property stating that the variance is the second central moment of a distribution.

Key concepts

Moment Generating Function

The Moment Generating Function, often abbreviated as the mgf, is a powerful tool in probability and statistics. It is similar to a fingerprint for a distribution, capturing essential information that fully describes it, if it exists for all values of the parameter.
To compute the mgf, we use the function \( M(t) = E[e^{tX}] \), where \(E\) denotes the expected value and \(X\) is the random variable.

• The mgf, when it exists, can be used to find moments of the distribution. This is why it is termed "moment generating function."
• The value of the mgf at zero, \( M(0) \), is always equal to 1 for any distribution.
• Each derivative of the mgf evaluated at zero gives us a specific moment about the origin. The first derivative gives the mean, while the second gives the variance and so on.

The existence of the moment generating function indicates that all moments of the probability distribution exist and are finite. It’s a cornerstone method in probability for finding the characteristics of distributions and proving statistical theorems.

Mean of the Distribution

The mean of a distribution is one of its central and most intuitive statistical measures. It is essentially the "average" value and gives a sense of where the bulk of the distribution is located.

• The mean can be calculated using the mgf by taking the first derivative \( M'(t) \) and evaluating it at \( t = 0 \).
• In terms of the cumulant generating function \( \psi(t) = \log M(t) \), the first derivative is \( \psi'(0) = \mu \), where \( \mu \) is the mean.
• Understanding where the mean lies helps in various statistical applications, from simple data analysis to complex predictive modeling.

Knowing the mean allows us to summarize the distribution with a single number that represents its center, providing crucial insight in interpreting statistical data.

Variance of the Distribution

Variance is another fundamental measure in statistics that complements the mean. It tells us how much the values in a distribution deviate from its mean, essentially providing a measure of spread or dispersion.

• The variance is calculated using the second derivative of the moment generating function, \( M''(t) \), also evaluated at \( t = 0 \).
• In the context of the cumulant generating function, this means \( \psi''(0) = \sigma^2 \), which is the variance.
• The concept of variance is essential when understanding how data points differ from each other and from their mean, impacting interpretations and predictions.

Variance is a key concept because it affects everything from the bell shape of a normal distribution to the uncertainty in estimating parameters from data. Understanding both the mean and variance gives a comprehensive view of a distribution's behavior.