Which property is unique to the MGF of a geometric distribution?

Which property is unique to the MGF of a geometric distribution?

• A. It is always increasing
• B. It is defined for all real numbers
• C. It can be used to derive variance and mean
• D. It converges for larger values of t

Answer: (C)

The moment-generating function (MGF) of a geometric distribution has several notable properties, but the ability to derive the mean and variance from it is particularly key. For the geometric distribution, the MGF is defined as \( M(t) = \frac{p e^t}{1 - (1-p)e^t} \) for \( t < -\ln(1-p) \). Through differentiation of the MGF with respect to \( t \), one can find the first moment, which corresponds to the mean, and the second moment, which allows for the calculation of variance. The first derivative of the MGF evaluated at \( t = 0 \) gives the mean, while the second derivative evaluated at \( t = 0 \) provides the second moment needed for variance calculations. This property of the MGF not only highlights its functional significance in probability theory, but also differentiates it in context. While other distributions have MGFs that can be used similarly, the MGF of a geometric distribution succinctly captures this relationship to the mean and variance in a straightforward manner. The other properties, such as being defined for all real numbers or converging for larger values of t, are not unique to the geometric distribution and

The moment-generating function (MGF) of a geometric distribution has several notable properties, but the ability to derive the mean and variance from it is particularly key.

For the geometric distribution, the MGF is defined as ( M(t) = \frac{p e^t}{1 - (1-p)e^t} ) for ( t < -\ln(1-p) ). Through differentiation of the MGF with respect to ( t ), one can find the first moment, which corresponds to the mean, and the second moment, which allows for the calculation of variance. The first derivative of the MGF evaluated at ( t = 0 ) gives the mean, while the second derivative evaluated at ( t = 0 ) provides the second moment needed for variance calculations.

This property of the MGF not only highlights its functional significance in probability theory, but also differentiates it in context. While other distributions have MGFs that can be used similarly, the MGF of a geometric distribution succinctly captures this relationship to the mean and variance in a straightforward manner.

The other properties, such as being defined for all real numbers or converging for larger values of t, are not unique to the geometric distribution and