chapter 6 stat

random variable

A variable that takes numerical values based on the outcomes of a random process.

probability distribution

A list or rule that shows all possible values of a random variable and their probabilities.

discrete random variable

A random variable that takes countable values, such as whole numbers.

mean (expected value) of a discrete random variable

The long-run average value found by multiplying each value by its probability and adding the results.

standard deviation of a discrete random variable

A measure of how much the values of a random variable typically vary from the mean.

variance

The square of the standard deviation, measuring overall spread of a random variable.

continuous random variable

A random variable that can take any value within an interval.

how to calculate probabilities for a continuous random variable

Probabilities are found by calculating the area under the density curve over an interval.

effect of adding or subtracting a constant on a probability distribution

Adding or subtracting a constant shifts the distribution and changes the mean but not the standard deviation.

effect of multiplying or dividing by a constant on a probability distribution

Multiplying or dividing by a constant changes both the mean and the standard deviation by that constant.

effect of a linear transformation on a random variable

A linear transformation changes the mean and standard deviation according to the formula a + bX.

mean (expected value) of a sum of random variables

The mean of a sum equals the sum of the individual means.

mean (expected value) of a difference of random variables

The mean of a difference equals the difference of the individual means.

independent random variables

Random variables are independent if knowing one does not affect the probability of the other.

standard deviation of the sum of two independent random variables

The standard deviation is the square root of the sum of their variances.

standard deviation of the difference of two independent random variables

The standard deviation is the square root of the sum of their variances.

mean and standard deviation of a linear combination of random variables

The mean follows linear rules, and the standard deviation depends on variances and independence.

combining normal random variables

The sum or difference of independent normal random variables is also normally distributed.

binomial setting

A situation with a fixed number of trials, two outcomes, constant probability, and independent trials.

binomial random variable

The number of successes in a fixed number of binomial trials.

binomial distribution

The probability distribution of a binomial random variable.

binomial coefficient

The number of ways to choose k successes from n trials, written as nCk.

binomial probability formula

A formula that calculates the probability of exactly k successes in n trials.

how to find binomial probabilities

Use the binomial formula or technology to calculate the probability of a given number of successes.

mean of a binomial random variable

The mean is equal to n times p.

standard deviation of a binomial random variable

The standard deviation is the square root of np(1 − p).

10% condition

Sampling without replacement is approximately independent if the sample is less than 10% of the population.

normal approximation for binomial distributions: the large counts condition

The normal approximation can be used if np ≥ 10 and n(1 − p) ≥ 10.

geometric setting

A situation where trials are independent, have two outcomes, and continue until the first success.

geometric random variable

The number of trials needed to get the first success.

geometric distribution

The probability distribution of a geometric random variable.

geometric probability formula

A formula that finds the probability that the first success occurs on the kth trial.

mean (expected value) of a geometric random variable

The mean is equal to 1 divided by p.

standard deviation of a geometric random variable

The standard deviation is √((1 − p)/p²).