Normal Distribution

A normal or Gaussian distribution is continuous, symmetric, unimodal, bell-shaped, and asymptotic* to the horizontal axis.
The mean, mode, and median of a normal distribution all coincide with the same point in the center of the distribution.
The shorthand notation for a normal distribution with mean $\mu$ and standard deviation $\sigma$ is $N(\mu, \sigma)$.

Standard Normal Distribution

The Standard Normal Distribution is a normal distribution with mean $\mu=0$ and standard deviation $\sigma=1$.

Let $X$ be a continuous random variable. If $X\sim N(\mu, \sigma)$, then $Z$ follows the Standard Normal Distribution. That is:

$$Z=\cfrac{X-\mu}{\sigma}\sim N(0,1)$$

Binomial Probability Distribution

Let $X$ be a binomially distributed random variable with parameters $n$ and $p$.
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To calculate the expected value of $X$, make use of the following formula:

$$\mu_{\small{X}} = \mathbb{E}[X]=n\cdot p$$
To calculate the variance of $X$, make use of the following formula:
$$\sigma^2_{\small{X}} = Var[X]=n\cdot p \cdot (1-p)$$
To calculate the standard deviation of $X$, make use of the following formula:
$$\sigma_{\small{X}} = SD[X]=\sqrt{n\cdot p \cdot (1-p)}$$

Sampling Distribution of the Sample Mean

If the conditions for normality are satisfied, the sampling distribution of the sample mean is approximately normally distributed with parameters:

• Expected value of the sample mean: $\mu_{\bar{X}} = \mu$
• Standard error of the sample mean: $\sigma_{\bar{X}} = \cfrac{\sigma}{\sqrt{n}}$

Sampling Distribution of the Sample Proportion

If the conditions for normality are satisfied, the sampling distribution of the sample proportion is approximately normally distributed with parameters:

• Expected value of the sample proportion: $\mu_{\hat{p}} = \pi$
• Standard error of the sample proportion: $\sigma_{\hat{p}} = \sqrt{\cfrac{\pi(1-\pi)}{n}}$