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In probability theory, the characteristic function of a random variable is a complex-valued function that describes the distribution of that variable. In LaTeX, the characteristic function can be represented using the command \varphi or \phi.
In probability theory, the characteristic function of a random variable is a complex-valued function that describes the distribution of that variable. In LaTeX, the characteristic function can be represented using the command \varphi or \phi.
To write the characteristic function in LaTeX, use the following command:
$$
\varphi_X(t) = \mathbb{E}[e^{itX}]
$$
$$ \varphi_X(t) = \mathbb{E}[e^{itX}] $$
This represents the characteristic function of a random variable X.
Here are some examples of using the \varphi command:
$$
\varphi_{\mathrm{Bin}(n,p)}(t) = (pe^{it} + 1 - p)^n
$$
$$ \varphi_{\mathrm{Bin}(n,p)}(t) = (pe^{it} + 1 - p)^n $$
This represents the characteristic function of a binomial distribution with parameters n and p.
$$
\varphi_{\mathcal{N}(\mu,\sigma^2)}(t) = e^{it\mu - \frac{1}{2}\sigma^2t^2}
$$
$$ \varphi_{\mathcal{N}(\mu,\sigma^2)}(t) = e^{it\mu - \frac{1}{2}\sigma^2t^2} $$
This represents the characteristic function of a normal distribution with mean $\mu$ and variance $\sigma^2$.
Remember that to use the \varphi command in LaTeX, you need to include the package amsmath in the preamble of your LaTeX document.
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