AP Statistics TPS4e Chapter 6 Random Variables Vocabulary

Geometric setting

A _____arises when we perform independent trials of the same chance process and record the number of trials until a particular outcome occurs. The four conditions for a ______ are: • Binary - The possible outcomes of each trial can be classified as "success" or "failure." • Independent - Trials must be independent; that is, knowing the result of one trial must not have any effect on the result of any other trial. • Trials - The goal is to count the number of trials until the first success occurs. • Success - On each trial, the probability p of success must be the same.

Linear transformation

A linear transformation of a random variable involves adding a constant a, multiplying by a constant b, or both. We can write a ______of the random variable X in the form Y = a + bX. The shape, center, and spread of the probability distribution of Y are as follows: Shape: Same as the probability distribution of X. Center: μY = a + b(μX) Spread: σY = |b|σX

Binomial setting

Arises when we perform several independent trials of the same chance process and record the number of times that a particular outcome occurs. The four conditions for a ______ are: • Binary - The possible outcomes of each trial can be classified as "success" or "failure." • Independent - Trials must be independent; that is, knowing the result of one trial must not have any effect on the result of any other trial. • Number - The number of trials n of the chance process must be fixed in advance. • Success - On each trial, the probability p of success must be the same.

Factorial

For any positive whole number n, its factorial n! is n! = n ∙ (n − 1) ∙ (n − 2) ∙ ... ∙ 3 ∙ 2 ∙ 1 In addition, we define 0! = 1.

Variance of the sum (difference) of independent random variables

For any two independent random variables X and Y, if T = X + Y, then the variance of T is σT Squared = σX Sqaured + σY Squared. If D = X - Y, then the variance of D is σD Squared = σX Squared + σY Squared.

Mean of the sum (difference) of random variables

For any two random variables X and Y, if T = X + Y then the mean of T is μT = μX + μY. If D = X - Y , then the mean of D is μD = μX - μY. In general, the mean of the sum (difference) of several random variables is the sum (difference) of their means.

Binomial probability

If X has the binomial distribution with n trials and probability p of success on each trial, the possible values of X are 0, 1, 2, ..., n.

Normal approximation for binomial distributions

If X is a count having the binomial distribution with parameters n and p, then when n is large, X is approximately Normally distributed with mean = np and standard deviation = Square root of np(1− p) . We will use this approximation when np ≥ 10 and n(1 - p) ≥ 10.

Geometric probability

If Y has the geometric distribution with probability p of success on each trial, the possible values of Y are 1, 2, 3, ....

Mean (expected value) of a geometric random variable

If Y is a geometric random variable with probability of success p on each trial, then its mean (expected value) is μY = E(Y) = 1/p . That is, the expected number of trials required to get the first success is 1/p.

Mean and standard deviation of a binomial random variable

If a count X has the binomial distribution with number of trials n and probability of success p, the mean and standard deviation of X are μX = np σX = Square root of np(1-p)

Independent random variables

If knowing whether any event involving X alone has occurred tells us nothing about the occurrence of any event involving Y alone, and vice versa, then X and Y are ______. That is, there is no association between the values of one variable and the values of the other.

Geometric distribution

In a _______, suppose we let Y = the number of trials required to get the first success. The probability distribution of Y is a _____ with parameter p, the probability of a success on any trial. The possible values of Y are 1, 2, 3, ....

Binomial distribution

In a binomial setting, suppose we let X = the number of successes. The probability distribution of X is a ______with parameters n and p, where n is the number of trials of the chance process and p is the probability of a success on any one trial. The possible values of X are the whole numbers from 0 to n.

Discrete random variable

Takes a fixed set of possible values with gaps between. The probability distribution of a ______ gives its possible values and their probabilities. The probability of any event is the sum of the probabilities for the values of the variable that make up the event.

Continuous random variable

Takes all values in an interval of numbers. The probability distribution of a ______ is described by a density curve. The probability of any event is the area under the density curve and above the values of the variable that make up the event.

Random Variable

Takes numerical values that describe the outcomes of some chance process.

Probability distribution

The _____ of a random variable gives its possible values and their probabilities.

Mean (expected value) of a random variable

The ______, X, denoted by μX , is the balance point of the probability distribution histogram or density curve. Since the mean is the long-run average value of the variable after many repetitions of the chance process, it is also known as the expected value E(X) of the random variable.

Variance of a random variable

The average squared deviation of the values of the variable from their mean. Also known as σX Squared

Binomial random variable

The count X of successes in a binomial setting.

Geometric random variable

The number of trials Y that it takes to get a success in a geometric setting.

Binomial coefficient

The number of ways of arranging k successes among n observations is given by the _____.

Standard deviation of a random variable

The square root of the variance of a random variable σX squared . The _______ measures the variability of the distribution about the mean.

Mean (expected value) of a discrete random variable X

To find the mean (expected value) of X, multiply each possible value by its probability, then add all the products: μX = E(X) = x1p1 + x2p2 + x3p3 + ... = ∑xi pi