AP Statistics Final Exam: Chapter 7 Random Variables

probability model for a continuous random variable

Assigns probabilities to intervals of outcomes rather than to individual outcomes. All continuous probability distributions assign probability 0 to every individual outcome. Only intervals of values have positive probability. The probability of any interval is the same as its length.

What is the relationship between normal distributions and probability distributions? Why?

Because any density curve describes an assignment of probabilities, Normal distributions are probability distributions.

probability distribution of a continuous random variable

Described by a density curve; the probability of any event is the area under the density curve and above the values of X that make up the event

law of large numbers

Draw independent observations at random from any population with finite mean u. Decide how accurately you would like to estimate u. As the number of observations drawn increases, the mean of the observed values eventually approaches the mean of the population as closely as you specified and then stays that close. *The average of the values of X observed in many trails must approach u.*

What should you do first when working these problems?

First identify the random variable of interest: X=number of _________ for discrete random variables, and X=amount of __________ for continuous random variables.

What is a standard Normal variable having the distribution N(0,1)?

If X has the N(u,o) distribution, then the standardized variable Z=X-u/o is a standard Normal random variable having the distribution N(0,1).

mean probability of any discrete random variable

It is an average of the possible outcomes, but a weighted average in which each outcome is weighted by its probability. Because the probabilities add to 1, we have total weight 1 to distribute among the outcomes. Suppose that X is a discrete random variable with a certain distribution. To find the mean of X, multiply each possible value by its probability, then add all the products.

sampling distributions

probability distributions of random variables

mean probability of a probability distribution

represented by u, or mu. Describes the long run average outcome; often called the expected value

probability histograms

used to show probability distribution as well as distribution of data; the height of each bar shows the probability of the outcome at its base. Because the heights are probabilities, they add up to 1. All the bars in a histogram have the same width as usual. Therefore, areas of the bars also display the assignment of probability to outcomes.

random variable

variable whose value is a numerical outcome of a random phenomenon; usually denoted by capital letters such as X or Y; Example: mean of a random sample

What are the two requirements which the probabilities pi must satisfy?

1) Every probability pi is a number between 0 and 1. 2) The sum of the probabilities is 1: p1 + p2 +...+ pk=1 You can find the probability of any event by adding the probabilities pi of the particular values xi, that make up the event.

continuous random variable

A continuous random variable X takes all values in an interval of numbers.

discrete random variable

A discrete random variable X has a countable number of possible values

probability distribution of a discrete random variable

The probability distribution of a discrete random variable X lists the values and their probabilities.

Mean of a set of observations

Their ordinary average in the set of observations. the mean of a discrete random variable X is also an average of the possible values of X, but with an essential change to take into account the fact that not all outcomes need be equally likely.

What are variance and standard deviations of random variables measures of?

They are measures of spread that accompany the choice of the mean to measure center. The standard deviation of a discrete random variable is the square root of the variance.

What is the purpose of probability histograms?

They make it easy to quickly compare two distributions.

What happens with the greater than and greater than or equal to signs when finding probabilities for continuous, but not discrete, random variables?

We ignore the distinction between > and >= signs when finding probabilities for continuous, but not discrete, random variables.