Illustrate how characteristic functions can be used to calculate moments of a random
variable.

### Characteristic Functions and Moments of a Random Variable

A **characteristic function** is a powerful tool in probability theory, closely related to the moment-generating function. It is defined for a random variable \(X\) as:

\[
\phi_X(t) = \mathbb{E}\left[e^{itX}\right],
\]

where \(t\) is a real number, \(i\) is the imaginary unit (\(i = \sqrt{-1}\)), and \(\mathbb{E}\) denotes the expectation.

### Moments from the Characteristic Function

The moments of a random variable \(X\) can be derived from the characteristic function by differentiating it with respect to \(t\) and then evaluating at \(t = 0\). Here's how it works:

1. **First Moment (Mean)**:
   The first moment or the mean \(\mu = \mathbb{E}[X]\) can be obtained by differentiating \(\phi_X(t)\) with respect to \(t\):

   \[
   \mu = \mathbb{E}[X] = \left. \frac{d\phi_X(t)}{dt} \right|_{t=0}.
   \]

   **Proof**:
   \[
   \frac{d\phi_X(t)}{dt} = \frac{d}{dt} \mathbb{E}[e^{itX}] = \mathbb{E}\left[\frac{d}{dt} e^{itX}\right] = \mathbb{E}\left[iX e^{itX}\right].
   \]
   Evaluating at \(t = 0\),
   \[
   \frac{d\phi_X(t)}{dt}\Big|_{t=0} = \mathbb{E}[iX] = i\mathbb{E}[X].
   \]
   Therefore, 
   \[
   \mu = \mathbb{E}[X] = \text{Im}\left(\frac{d\phi_X(t)}{dt}\Big|_{t=0}\right).
   \]

2. **Second Moment (Variance)**:
   The second moment is related to the variance \(\sigma^2 = \mathbb{E}[(X - \mu)^2]\). We can obtain it from the second derivative of the characteristic function:

   \[
   \mathbb{E}[X^2] = -\left.\frac{d^2\phi_X(t)}{dt^2}\right|_{t=0}.
   \]

   **Proof**:
   \[
   \frac{d^2\phi_X(t)}{dt^2} = \frac{d}{dt} \mathbb{E}[iXe^{itX}] = \mathbb{E}\left[\frac{d}{dt} (iXe^{itX})\right] = \mathbb{E}\left[-X^2 e^{itX}\right].
   \]
   Evaluating at \(t = 0\),
   \[
   \frac{d^2\phi_X(t)}{dt^2}\Big|_{t=0} = -\mathbb{E}[X^2].
   \]

   Therefore, 
   \[
   \text{Variance } \sigma^2 = \mathbb{E}[X^2] - (\mathbb{E}[X])^2.
   \]

3. **Higher Moments**:
   In general, the \(n\)-th moment of \(X\) can be obtained from the \(n\)-th derivative of the characteristic function:

   \[
   \mathbb{E}[X^n] = \left. \frac{d^n\phi_X(t)}{dt^n} \right|_{t=0} \cdot \frac{1}{i^n}.
   \]

### Summary

The characteristic function is a crucial tool in probability theory, as it uniquely determines the distribution of a random variable. By differentiating the characteristic function with respect to \(t\) and evaluating it at \(t=0\), one can calculate all the moments (mean, variance, skewness, kurtosis, etc.) of the random variable.

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Would you like further details or have any questions?

### 5 Related Questions

1. How does the moment-generating function compare to the characteristic function?
2. Can characteristic functions be used to identify the distribution of a random variable?
3. How are characteristic functions used in the context of the Central Limit Theorem?
4. What is the relationship between the Fourier transform and characteristic functions?
5. How do you compute the skewness and kurtosis of a random variable using characteristic functions?

### Tip

When working with characteristic functions, remember that they are always well-defined, even when the moment-generating function might not exist. This makes them particularly useful for analyzing distributions with heavy tails or undefined moments.

Learn how characteristic functions can be used to calculate moments of a random variable in probability theory. Understand the definitions and formulas involved, including mean, variance, and higher moments. Suitable for advanced undergraduate students and professionals in statistics and probability.

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