To derive the characteristic function of $X^2$, we first need to understand what you're asking. Let's break it down.

1. What is a characteristic function? The characteristic function $\varphi_X(t)$ of a random variable $X$ is defined as the expected value of $e^{itX}$, where $i$ is the imaginary unit and $t$ is a real parameter. Mathematically: $\varphi_X(t) = \mathbb{E}[e^{itX}]$ In other words, it's the Fourier transform of the probability distribution of $X$.

2. What is $X^2$? You are asking for the characteristic function of $X^2$, where $X \sim N(0, 1)$, meaning $X$ is a normally distributed random variable with mean 0 and variance 1.

Now, let's derive the characteristic function of $X^2$.

Step 1: Characteristic function of $X^2$

We want to find the characteristic function of $Y = X^2$, where $X \sim N(0, 1)$. That is, we need to compute: $\varphi_Y(t) = \mathbb{E}[e^{itX^2}]$ where $Y = X^2$.

Step 2: Use the PDF of $X$

Since $X$ is normally distributed with mean 0 and variance 1, its probability density function (PDF) is given by: $f_X(x) = \frac{1}{\sqrt{2\pi}} e^{-x^2 / 2}$

Step 3: Set up the integral for the characteristic function

We now compute the expectation: $\varphi_Y(t) = \mathbb{E}[e^{itX^2}] = \int_{-\infty}^{\infty} e^{itx^2} f_X(x) \, dx$ Substitute $f_X(x)$: $\varphi_Y(t) = \int_{-\infty}^{\infty} e^{itx^2} \frac{1}{\sqrt{2\pi}} e^{-x^2 / 2} \, dx$ Simplify the exponent: $\varphi_Y(t) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} e^{-(1/2 - it)x^2} \, dx$

Step 4: Solve the Gaussian integral

The above integral is a Gaussian integral of the form: $\int_{-\infty}^{\infty} e^{-ax^2} \, dx = \sqrt{\frac{\pi}{a}} \quad \text{for} \ a > 0$ Here, $a = \frac{1}{2} - it$. Therefore, the characteristic function becomes: $\varphi_Y(t) = \frac{1}{\sqrt{2\pi}} \cdot \sqrt{\frac{\pi}{\frac{1}{2} - it}}$ Simplify this expression: $\varphi_Y(t) = \frac{1}{\sqrt{1 - 2it}}$

Thus, the characteristic function of $Y = X^2$ is: $\varphi_Y(t) = \frac{1}{\sqrt{1 - 2it}}$

Conclusion:

The characteristic function of $X^2$, where $X \sim N(0,1)$, is: $\varphi_{X^2}(t) = \frac{1}{\sqrt{1 - 2it}}$

Would you like a more detailed breakdown or have any other questions?

Here are 5 related questions:

1. How do we compute the characteristic function for other distributions, such as exponential or uniform?
2. What is the relationship between the characteristic function and the moment-generating function?
3. Can we use the inverse Fourier transform to retrieve the PDF from the characteristic function?
4. How does the characteristic function change if $X$ has a nonzero mean or a different variance?
5. How can we use the characteristic function to find moments of $X^2$?

Tip: The characteristic function is useful because it uniquely defines the distribution and simplifies the computation of moments and other properties of random variables.