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, X¯X¯, and the sample sum, ΣΧ

 
If the size (n) of the sample is sufficiently large, then X¯X¯ ~ N(μ, σnσn) and ΣΧ ~ N(, (nn)(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, σnσn, 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 = 1m1m and the standard deviation is σ = 1m1m. The probability density function is f(x) = memx, x ≥ 0, and the cumulative distribution function is P(Xx) = 1 – emx.
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 x¯x¯) is x¯ = sum of all values in the samplenumber of values in the samplex¯ = sum of all values in the samplenumber of values in the sample, and the mean for a population (denoted by μ) is μ = sum of all values in the populationnumber of values in the populationμ = sum of all values in the populationnumber of values in the population.
normal distribution
a continuous random variable (RV) with probability density function (pdf) f(x) = 1σ2π e(x  μ)22σ2f(x) = 1σ2π e(x  μ)22σ2, 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 σnσn
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 rectangle

 
Notation: X ~ U(a, b). The mean is μ = a + b2μ = a + b2 and the standard deviation is σ = (ba)212σ = (ba)212. The probability density function is f(x) = 1baf(x) = 1ba for a < x < b or axb. The cumulative distribution is P(Xx) = xabaxaba.