What's this mathematical symbol?

Question

What does the mathematical symbol with "an I inside the O" mean?

Answer 1

$\Phi$ and $\phi$ are two standardized symbols to get to know well, whenever you're reading anything on probability.

$\Phi$ is the cumulative distribution function of the standard normal distribution; i.e., the normal distribution with mean $0$ and variance $1$. $\phi$ is the corresponding (probability) density function to $\Phi$. Suppose $Z$ is a standard normal random variable. Then for $z \in \mathbb{R}$,

$$\Phi(z) = \mathbb{P}(Z\leq z) = F_{Z}(z) = \int_{-\infty}^{z}f_{Z}(t)\text{ d}t=\int_{-\infty}^{z}\phi(t)\text{ d}t$$ - that is, $\Phi = F_{Z}$ and $\phi = f_{Z}$.

In general, suppose $X$ is a normally-distributed random variable with mean $\mu$ and variance $\sigma^2$, with $\sigma = \sqrt{\sigma^2}$. Then for $x \in \mathbb{R}$, $$\mathbb{P}(X \leq x) = \mathbb{P}\left(\dfrac{X - \mu}{\sigma} \leq \dfrac{x - \mu}{\sigma}\right) = \mathbb{P}\left(Z \leq \dfrac{x-\mu}{\sigma}\right)=\Phi\left(\dfrac{x-\mu}{\sigma}\right)\text{.}$$

Answer 2

$\Phi$ is the Greek symbol Phi denoted by $\Phi $. You can see the Wiki page for more information.

In probability theory, $\Phi_{x} = \frac {1}{\sqrt {2\pi}} e^{-\frac {x^2}{2}}$ is the probability density function of the normal distribution.

Hope it helps.

Answer 3

In probability, $\Theta$ represents the parameter space. For example, for the normal distribution, the parameters are $\mu$ and $\sigma$ where $\mu \in \mathbb{R}$ and $\sigma \in [0,\infty)$, so the parameter space $\Theta$ is \begin{align*} \Theta = \{(\mu,\sigma) : \mu \in \mathbb{R}, \sigma \in [0,\infty)\} \end{align*}

$\Phi$ typically represents the cumulative distribution function. So if the distribution is characterized by the density $f$, then \begin{align*} \Phi(x) = \int_{-\infty}^x f(t) dt \end{align*}

In the discrete case, the integral is replaced by a sum, and $f(t)$ is replaced by the pmf.