Moment-Generating Function question

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In summary, a moment-generating function (MGF) is a mathematical function that fully describes the probability distribution of a random variable. It differs from a probability distribution function (PDF) in that it provides information about the entire distribution rather than specific outcomes. The MGF can be calculated by taking the expected value of the exponential function raised to the power of the random variable, and can provide information about moments, probabilities, and distribution type. However, MGFs have limitations such as only being applicable to random variables with defined mean and variance, difficulty in complex distributions, and a lack of information about extreme values.
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Q:Explain why there can be no random variable for which $M_x(t)=\frac{t}{t-1}$
($M_x(t)$ is the moment-generating function of the random variable $x$.)

A: I tried differentiating it twice and got mean of $x=1$ and variance of $x=1$ which seems fine. Maybe its because $M_x(0)$ is not equal to $1$?Thanks in advance for any help with this problem.
 
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JGalway said:
Maybe its because $M_x(0)$ is not equal to $1$?

Yes, that's exactly the reason.
 

1. What is a moment-generating function (MGF)?

An MGF is a mathematical function that is used to fully describe the probability distribution of a random variable. It is defined as the expected value of the exponential function raised to the power of the random variable.

2. How is an MGF different from a probability distribution function (PDF)?

An MGF is a mathematical function that provides information about the entire probability distribution of a random variable, while a PDF only describes the probabilities of specific outcomes. MGFs are also more useful for calculating moments (such as mean and variance) of a distribution.

3. How do I calculate the MGF of a random variable?

The MGF of a random variable can be calculated by taking the expected value of the exponential function raised to the power of the random variable. This can be done using calculus or by finding the moments of the distribution and using them to construct the MGF.

4. What information can be obtained from an MGF?

An MGF provides information about the moments of a distribution, such as mean and variance. It can also be used to find the probability of a range of values for a random variable and to calculate the probabilities of different outcomes for a combination of random variables. Additionally, MGFs can be used to determine the type of probability distribution a random variable follows.

5. Are there any limitations to using MGFs?

MGFs can only be used for random variables that have a defined mean and variance. They also may not exist for certain distributions or may be difficult to calculate for complex distributions. Additionally, MGFs may not provide information about the tails of a distribution, making them less useful for extreme values.

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