Chapter 5 statistics

Continuous Random Variable

A quantitative variable is continuous if its set of possible values is uncountable. Examples include temperature, exact height, exact age (including parts of a second). In practice, one can never measure a continuous variable to infinite precision, so continuous variables are sometimes approximated by discrete variables. A random variable X is also called continuous if its set of possible values is uncountable, and the chance that it takes any particular value is zero (in symbols, if P(X = x) = 0 for every real number x). A random variable is continuous if and only if its cumulative probability distribution function is a continuous function (a function with no jumps).

Discrete Random Variable

A quantitative variable whose set of possible values is countable. Typical examples of discrete variables are variables whose possible values are a subset of the integers, such as Social Security numbers, the number of people in a family, ages rounded to the nearest year, etc. Discrete variables are "chunky." C.f. continuous variable. A discrete random variable is one whose set of possible values is countable. A random variable is discrete if and only if its cumulative probability distribution function is a stair-step function; i.e., if it is piecewise constant and only increases by jumps.

Normal Distribution

A random variable X has a normal distribution with mean m and standard error s if for every pair of numbers a ≤ b, the chance that a < (X−m)/s < b is P(a < (X−m)/s < b) = area under the normal curve between a and b.

Binomial Distribution

A random variable has a binomial distribution (with parameters n and p) if it is the number of "successes" in a fixed number n of independent random trials, all of which have the same probability p of resulting in "success." Under these assumptions, the probability of k successes (and n−k failures) is nCk pk(1−p)n−k, where nCk is the number of combinations of n objects taken k at a time: nCk = n!/(k!(n−k)!). The expected value of a random variable with the Binomial distribution is n×p, and the standard error of a random variable with the Binomial distribution is (n×p×(1 − p))½. This page shows the probability histogram of the binomial distribution.

Random Variable

A random variable is an assignment of numbers to possible outcomes of a random experiment. For example, consider tossing three coins. The number of heads showing when the coins land is a random variable: it assigns the number 0 to the outcome {T, T, T}, the number 1 to the outcome {T, T, H}, the number 2 to the outcome {T, H, H}, and the number 3 to the outcome {H, H, H}.

Expected Value

The expected value of a random variable is the long-term limiting average of its values in independent repeated experiments. The expected value of the random variable X is denoted EX or E(X). For a discrete random variable (one that has a countable number of possible values) the expected value is the weighted average of its possible values, where the weight assigned to each possible value is the chance that the random variable takes that value. One can think of the expected value of a random variable as the point at which its probability histogram would balance, if it were cut out of a uniform material. Taking the expected value is a linear operation: if X and Y are two random variables, the expected value of their sum is the sum of their expected values (E(X+Y) = E(X) + E(Y)), and the expected value of a constant a times a random variable X is the constant times the expected value of X (E(a×X ) = a× E(X)).

Probability Distribution

The probability distribution of a random variable specifies the chance that the variable takes a value in any subset of the real numbers. (The subsets have to satisfy some technical conditions that are not important for this course.) The probability distribution of a random variable is completely characterized by the cumulative probability distribution function; the terms sometimes are used synonymously. The probability distribution of a discrete random variable can be characterized by the chance that the random variable takes each of its possible values. For example, the probability distribution of the total number of spots S showing on the roll of two fair dice can be written as a table:

Standard Deviation

The standard deviation of a set of numbers is the rms of the set of deviations between each element of the set and the mean of the set. See also sample standard deviation.

Binomial Random Variable

Variable may be defined as the number of successes in a given number of trials where the outcome can be either a success or failure; Expected value = (probability of success) * (number of trials); Variance = (expected value) * (1 - probability of success)