Download Big Idea Terms for Chapter 4

Big Idea Terms for Chapter 4
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.
Central Limit Theorem.
The central limit theorem states that the PROBABILITY HISTOGRAMS of the SAMPLE MEAN and
SAMPLE SUM of n draws with replacement from a box of labeled tickets converge to a NORMAL
CURVE as the SAMPLE SIZE n grows, in the following sense: As n grows, the area of the
probability histogram for any range of values approaches the area under the NORMAL CURVE for
the same range of values, converted to STANDARD UNITS. See also THE NORMAL
APPROXIMATION.
Expected value is one of the fundamental concepts in probability, in a sense more general than
probability itself. The expected value of a real-valued random variable gives a measure of the
center of the distribution of the variable.
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.
If there are numbers m and s such that X has a normal distribution with mean m and standard
error s, then X is said to have a normal distribution or to be normally distributed. If X has a
normal distribution with mean m=0 and standard error s=1, then X is said to have a standard
normal distribution. The notation X~N(m,s2) means that X has a normal distribution with mean
m and standard error s; for example, X~N(0,1), means X has a standard normal distribution.
Probability density function.
The chance that a CONTINUOUS RANDOM VARIABLE is in any range of values can be calculated
as the area under a curve over that range of values. The curve is the probability density
function of the random variable. That is, if X is a continuous random variable, there is a function
f(x) such that for every pair of numbers a≤b,
P(a≤ X ≤b) = (area under f between a and b);
f is the probability density function of X. For example, the probability density function of a
random variable with a STANDARD NORMAL DISTRIBUTION is the NORMAL CURVE. Only
continuous random variables have probability density functions.
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:
s P(S=s)
2 1/36
3 2/36
4 3/36
5 4/36
6 5/36
7 6/36
8 5/36
9 4/36
10 3/36
11 2/36
12 1/36
The probability distribution of a CONTINUOUS RANDOM VARIABLE can be characterized by its
PROBABILITY DENSITY FUNCTION.
Standard Error (SE).
The Standard Error of a RANDOM VARIABLE is a measure of how far it is likely to be from its
EXPECTED VALUE; that is, its scatter in repeated experiments. The SE of a random variable X is
defined to be
SE(X) = [E( (X − E(X))2 )] ½.
That is, the standard error is the square-root of the EXPECTED squared difference between the
random variable and its expected value. The SE of a random variable is analogous to the SD of
a list.