Determine the first two moments of an Exponential random variable using the characteristic 
function of the exponential distribution. 

To determine the first two moments (mean and variance) of an Exponential random variable using its characteristic function, we start by understanding the definition and properties of the exponential distribution.

### Step 1: Exponential Distribution and Its Characteristic Function

An exponential random variable \( X \) with rate parameter \( \lambda > 0 \) has the probability density function (PDF):

\[
f_X(x) = \lambda e^{-\lambda x}, \quad x \geq 0.
\]

The characteristic function \( \phi_X(t) \) of a random variable \( X \) is given by:

\[
\phi_X(t) = \mathbb{E}[e^{itX}] = \int_{-\infty}^{\infty} e^{itx} f_X(x) \, dx.
\]

Substitute the PDF of the exponential distribution:

\[
\phi_X(t) = \int_{0}^{\infty} e^{itx} \lambda e^{-\lambda x} \, dx.
\]

Simplify the exponent:

\[
\phi_X(t) = \lambda \int_{0}^{\infty} e^{(it - \lambda) x} \, dx.
\]

### Step 2: Evaluate the Characteristic Function

To evaluate this integral, notice that it is of the form:

\[
\int_{0}^{\infty} e^{(a)x} \, dx = \left[ \frac{e^{ax}}{a} \right]_{0}^{\infty} \text{ for } a < 0.
\]

Here, \( a = it - \lambda \). For convergence, we need \( \text{Re}(\lambda - it) > 0 \), which is true since \( \lambda > 0 \). So, we have:

\[
\phi_X(t) = \lambda \left( \frac{1}{it - \lambda} \right) = \frac{\lambda}{\lambda - it}.
\]

### Step 3: Compute the Moments Using the Characteristic Function

The \( n \)-th moment of a random variable \( X \) can be obtained by differentiating its characteristic function \( \phi_X(t) \) with respect to \( t \) and then evaluating at \( t = 0 \):

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

#### First Moment (Mean)

To find the first moment (mean), differentiate \( \phi_X(t) \) once with respect to \( t \):

\[
\frac{d}{dt} \phi_X(t) = \frac{d}{dt} \left( \frac{\lambda}{\lambda - it} \right).
\]

Using the quotient rule:

\[
\frac{d}{dt} \phi_X(t) = \frac{0 \cdot (\lambda - it) - \lambda (-i)}{(\lambda - it)^2} = \frac{i\lambda}{(\lambda - it)^2}.
\]

Evaluate at \( t = 0 \):

\[
\left. \frac{d}{dt} \phi_X(t) \right|_{t=0} = \frac{i \lambda}{\lambda^2} = \frac{i}{\lambda}.
\]

The mean \( \mathbb{E}[X] \) is the imaginary part divided by \( i \):

\[
\mathbb{E}[X] = \frac{1}{\lambda}.
\]

#### Second Moment

Now, find the second moment by differentiating \( \phi_X(t) \) twice with respect to \( t \):

\[
\frac{d^2}{dt^2} \phi_X(t) = \frac{d}{dt} \left( \frac{i \lambda}{(\lambda - it)^2} \right).
\]

Apply the quotient rule again:

\[
\frac{d^2}{dt^2} \phi_X(t) = \frac{(i\lambda)(2i)(\lambda - it)}{(\lambda - it)^4} = \frac{-2\lambda}{(\lambda - it)^3}.
\]

Evaluate at \( t = 0 \):

\[
\left. \frac{d^2}{dt^2} \phi_X(t) \right|_{t=0} = \frac{-2\lambda}{\lambda^3} = -\frac{2}{\lambda^2}.
\]

The second moment \( \mathbb{E}[X^2] \) is:

\[
\mathbb{E}[X^2] = \frac{2}{\lambda^2}.
\]

#### Variance

The variance \( \text{Var}(X) \) is:

\[
\text{Var}(X) = \mathbb{E}[X^2] - (\mathbb{E}[X])^2 = \frac{2}{\lambda^2} - \left( \frac{1}{\lambda} \right)^2 = \frac{2}{\lambda^2} - \frac{1}{\lambda^2} = \frac{1}{\lambda^2}.
\]

### Conclusion

The first two moments of the Exponential random variable \( X \) are:

1. **Mean (First Moment):** \(\mathbb{E}[X] = \frac{1}{\lambda}\).
2. **Variance (Second Moment):** \(\text{Var}(X) = \frac{1}{\lambda^2}\).

Would you like further details or have any questions?

---

Here are 5 related questions to explore:

1. How do we find moments using the moment-generating function (MGF)?
2. What are the properties of the exponential distribution that distinguish it from other distributions?
3. How does the exponential distribution relate to the Poisson process?
4. Can we use the exponential distribution to model real-world phenomena?
5. How do we extend the results to a gamma distribution, which generalizes the exponential distribution?

**Tip:** The characteristic function approach is a powerful tool for deriving moments, especially when dealing with complex distributions!

Learn how to determine the first two moments (mean and variance) of an Exponential random variable using its characteristic function. Detailed steps include deriving the characteristic function, computing moments, and understanding the relationship with the rate parameter. Suitable for advanced undergraduate students and researchers.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation