# Probability- finite n-th moment

• Roni1985
In summary, the conversation discusses the finite exponential moment of a random variable X and how it can be shown to have a finite nth moment for all positive integers n by comparing it to the Taylor series for EXP[x]. The individual mentions attempting to use the Taylor expansion but is unsure how to proceed. They express urgency for help as they have an upcoming exam in two days.
Roni1985

## Homework Statement

Suppose the random variable X has finite exponential moment. Show by comparison to the Taylor series for EXP[x] that X has finite nth moment (E|X|n<inf) for all positive integers n

## Homework Equations

ex=$$\sum$$($$\frac{x^n}{(n!)}$$, n,0,inf)

## The Attempt at a Solution

we know that E(et*x) < inf

I can use the Taylor expansion but I end up with

E(et*x) < $$\sum$$($$\frac{x^n}{(n!)}$$

don't know how to pick up from here...

would appreciate any help.

Thanks.

sorry for the bump, it's really important for me :\ have an exam in two days and I've been trying to solve it since 12 pm.

Thanks.

## 1. What is the definition of "finite n-th moment" in probability?

The finite n-th moment in probability refers to the expected value of a random variable raised to the n-th power, where the value of n is a positive integer. It is used to measure the central tendency and dispersion of a probability distribution.

## 2. How is the finite n-th moment calculated?

The finite n-th moment is calculated by taking the sum of all possible values of a random variable raised to the n-th power, weighted by their corresponding probabilities. This can also be expressed as an integral for continuous distributions.

## 3. What is the significance of the finite n-th moment in probability?

The finite n-th moment is significant in probability because it provides a mathematical measure of the spread and shape of a probability distribution. It can also be used to calculate other important measures such as variance and skewness.

## 4. How does the value of n affect the finite n-th moment?

The value of n affects the finite n-th moment by determining the order or degree of the moment. As n increases, the moment becomes more sensitive to extreme values, leading to a larger spread and higher order moments.

## 5. What are some real-life applications of the finite n-th moment?

The finite n-th moment has various applications in fields such as finance, engineering, and physics. It is particularly useful in risk assessment, portfolio management, and analyzing the stability of systems. It is also used in statistical models for data analysis and prediction.

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