Question

If M(t), t ∈ \(\R\), is the moment generating function of a random variable, then which one of the following is NOT the moment generating function of any random variable ?

• A. \(\frac{5e^{-5t}}{1-4t^2}M(t),|t| < \frac{1}{2}\)
• B. e^-tM(t), t ∈ \(\R\)
• C. \(\frac{1+e^t}{2(2-e^t)}M(t), t < \ln2\)
• D. M(4t), t ∈ \(\R\)

Answer 1

The Correct Option is A

To determine which given function is NOT a moment-generating function (MGF) of any random variable, we need to understand the properties of MGFs:

• An MGF, denoted by \(M(t)\), exists for some random variable if it is finite for all \(t\) in some open interval containing 0.
• MGFs are used to uniquely determine the distribution of the random variable, provided the MGF exists in a neighborhood of zero.
• MGFs must be analytic, meaning they must be expressible as a power series and defined in an interval around \(t = 0\).

Let's evaluate the given options:

1. \(\frac{5e^{-5t}}{1-4t^2}M(t), |t| < \frac{1}{2}\)
2. \(e^{-t}M(t), t \in \R\)
3. \(\frac{1+e^t}{2(2-e^t)}M(t), t < \ln2\)
4. \(M(4t), t \in \R\)

Consider each option:

• Option 1: \(\frac{5e^{-5t}}{1-4t^2}M(t)\)
  • The term \(\frac{1}{1-4t^2}\) suggests a form similar to a geometric series, which becomes infinite at \(t = \pm \frac{1}{2}\). Since MGFs need to be finite in a neighborhood of zero, this function cannot be an MGF because it becomes infinite within the valid region \(|t| < \frac{1}{2}\).
• Option 2: \(e^{-t}M(t)\)
  • Multiplying an MGF by an exponential function \(e^{-t}\) retains the analyticity and bounded nature of the MGF near \(t = 0\), assuming \(M(t)\) is valid for all \(t \in \R\). This is possible.
• Option 3: \(\frac{1+e^t}{2(2-e^t)}M(t)\)
  • This function involves the term \(\frac{1}{2-e^t}\). The denominator is zero at \(t = \ln 2\), hence it is undefined at this point. However, the function can still be finite in the given region \(t < \ln2\).
• Option 4: \(M(4t)\)
  • Rescaling the argument of an MGF retains its properties as an MGF, so long as \(M(t)\) is valid for the rescaled domain. This can still be an MGF.

Based on the analysis, the correct answer is:

\(\frac{5e^{-5t}}{1-4t^2}M(t), |t| < \frac{1}{2}\) because it is not finite at all points within its stipulated domain, making it unsuitable as an MGF.