Sections

Key Terms
# Key Terms

- average
- a number that describes the central tendency of the data; there are a number of specialized averages, including the arithmetic mean, weighted mean, median, mode, and geometric mean

- central limit theorem
- given a random variable (RV) with a known mean,
*μ*, and known standard deviation,*σ*, and sampling with size*n*, we are interested in two new RVs: the sample mean, $\overline{X}$, and the sample sum,*ΣΧ*If the size (

*n*) of the sample is sufficiently large, then $\overline{X}$ ~*N*(*μ*, $\frac{\sigma}{\sqrt{n}}$) and*ΣΧ*~*N*(*nμ*, ($\sqrt{n}$)(*σ*)). If the size (*n*) of the sample is sufficiently large, then the distribution of the sample means and the distribution of the sample sums will approximate a normal distribution regardless of the shape of the population. The mean of the sample means will equal the population mean, and the mean of the sample sums will equal*n*times the population mean. The standard deviation of the distribution of the sample means, $\frac{\sigma}{\sqrt{n}}$, is called the standard error of the mean.

- exponential distribution
- a continuous random variable (RV) that appears when we are interested in the intervals of time between a random events; for example, the length of time between emergency arrivals at a hospital, notation:
*X*~*Exp*(*m*)The mean is

*μ*= $\frac{1}{m}$ and the standard deviation is*σ*= $\frac{1}{m}$. The probability density function is*f*(*x*) =*me*^{–mx},*x*≥ 0, and the cumulative distribution function is*P*(*X*≤*x*) = 1 –*e*^{–mx}.

- mean
- a number that measures the central tendency; a common name for mean is
*average*; the term*mean*is a shortened form of*arithmetic mean*;by definition, the mean for a sample (denoted by $\overline{x}$) is $\overline{x}\text{=}\frac{\text{sumofallvaluesinthesample}}{\text{numberofvaluesinthesample}}$, and the mean for a population (denoted by

*μ*) is $\mu \text{=}\frac{\text{sumofallvaluesinthepopulation}}{\text{numberofvaluesinthepopulation}}$.

- normal distribution
- a continuous random variable (RV) with probability density function (pdf) $f(x)\text{=}\frac{1}{\sigma \sqrt{2\pi}}{e}^{\frac{\u2013{\text{(}x\text{}\u2013\text{}\mu )}^{2}}{2{\sigma}^{2}}}$, where
*μ*is the mean of the distribution and*σ*is the standard deviation; notation:*Χ*~*N*(*μ*,*σ*). If*μ*= 0 and*σ*= 1, the RV is called a**standard normal distribution**

- sampling distribution
- given simple random samples of size
*n*from a given population with a measured characteristic such as mean, proportion, or standard deviation for each sample, the probability distribution of all the measured characteristics is called a sampling distribution

- standard error of the mean
- the standard deviation of the distribution of the sample means, or $\frac{\sigma}{\sqrt{n}}$

- uniform distribution
- a continuous random variable (RV) that has equally likely outcomes over the domain
*a*<*x*<*b*; often referred as the**rectangular distribution**because the graph of the pdf has the form of a rectangleNotation:

*X*~*U*(*a*,*b*). The mean is $\mu \text{=}\frac{a\text{+}b}{2}$ and the standard deviation is $\sigma \text{=}\sqrt{\frac{{(b\u2013\text{a)}}^{2}}{12}}$. The probability density function is $f(x)\text{=}\frac{1}{b\u2013a}$ for*a*<*x*<*b*or*a*≤*x*≤*b*. The cumulative distribution is*P*(*X*≤*x*) = $\frac{x\u2013\text{a}}{b\u2013\text{a}}$.