The uniform distribution has a "flat" density function and thus a uniform probability in a closed interval $[A,B]$.

The density function of the continuous uniform random variable $X$ on the interval $[A,B]$ is

The mean and variance of the uniform distributions are

$$\mu =\frac{A+B}{2},{\sigma }^2=\frac{(B-A{)}^2}{12}.$$

The cumulative distribution function for a uniform distribution is

$$F(x)=\frac{x-A}{B-A},A\le x\le B$$

and the probability that a value falls between $a$ and $b$ is

$${\int }_a^b\frac{1}{B-A}dx=\frac{b-a}{B-A},A\le a,b\le B$$

The normal distribution or Gaussian distribution is the classic bell-shaped distribution.

The density of the normal random variable $X,$ with mean $\mu$ and variance ${\sigma }^2,$ is

The mean and variance of $n(x;\mu ,\sigma )$ are $\mu$ and ${\sigma }^2,$ respectively.

The standard normal distribution is the distribution of a normal random variable with mean $0$ and variance $1.$ We can transform a normal random variable $X$ into a standard normal random variable by

$$Z=\frac{X-\mu }{\sigma }.$$