AP Statistics Unit 4: Probability, Random variables, and Probability Distributions

What is a random process?

A random process generates outcomes that are determined purely by chance.

What is probability?

A number between 0 and 1 that describes the proportion of times an outcome will occur in a very long series of trials.

What is the law of large numbers?

If we observe more and more trials of any random process, the proportion of times that a specific outcome occurs approaches its probability. (LONG TERM)

What is a simulation?

An imitation of a random process in such a way that simulated outcomes are consistent with real-world outcomes.

How is a simulation performed?

1) Describe how to use a random process to do one trial of the simulation. Describe what is recorded after each trial.

2) Perform many trials of the simulation

3) Use the simulation results to answer the question of interest

What is a probability model?

The description of some random process that has two parts: a list of all possible outcomes and the probability of each outcome

The list of all possible outcomes is called a sample space

What is an event?

Any collection of outcomes from some random process.

What is the complement rule?

P(A^c) = 1 - P(A)

What are mutually exclusive events?

events that cannot happen at the same time

P(A and B) = 0

What is the addition rule for mutually exclusive events?

P(A or B) = P(A) + P(B)

What is the general addition rule?

P(A or B) = P(A) + P(B) - P(A and B)

What is a conditional probability?

Probability that one event happens given that another event is known to have already occurred.

P (A | B) means probability of A given B has already happened

P(A | B) = P(A and B) / P(B)

What are independent events?

Events where knowing whether or not one event has occurred does not change the probability that the other event will happen.

For example, if P(A | B) = P(A) because knowing event B has occurred does not change probability of A occurring

What is the general multiplication rule?

P(A and B) = P(A) x P(B | A)

What is a tree diagram?

A tree diagram shows the sample space of a random process with multiple stages. All probabilities after the first stage are conditional probabilities.

What is the multiplication rule for independent events?

P(A and B) = P(A) x P(B)

What is a random variable?

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

What is a probability distribution of a random variable?

Gives the possible values of a random variables and its probabilities.

What is a discrete random variable?

A variable that takes a fixed set of possible values with gaps between them.

What is expected value of a discrete random variable?

The mean of a discrete random variable. It is the average over many trials of the same random process.

E xipi

What is the standard deviation of a discrete random variable?

how much the values of the variable typically vary from the expected value in many trials of the random process.

The variance is: E(xi-mu(x))^2pi

What is a continuous random variable?

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

What is the effect of a linear transformation of a random variable?

let Y = a + bX be the linear transformation of the random variable X

mu(Y) = a + b(mu(x))

sigma(Y) = | b | x sigma(X)

What are independent random variables?

When knowing the value of one random variable doesn't change the probability distribution of the other random variable.

What is the mean and standard deviation of the sum of two independent random variables?

mu(X+Y) = mu(X) + mu(Y)

mu(X-Y) = mu(X) - mu(Y)

sigma(X+Y) = sqrt(sigma(X)^2 + sigma(Y)^2)

sigma(X-Y) = sqrt(sigma(X)^2 + sigma(Y)^2)

What is a binomial setting?

When we perform n independent trials of the same random process and count the number of times that a particular outcome occurs.

Conditions:

Binary: The possible outcomes must be success or failure.

Independent: Each trial must be independent.

Number: Fixed number of trials in advance

Same probability: Same P(success) in each trial.

What is a binomial random variable?

The binomial random variable is the count of number of successes in n trials.

What is a binomial distribution?

THe probability distribution of a binomial random variable. It is specified by number of trials n and probability p.

What is the binomial coefficient?

The number of ways to arrange x successes among n trials.

What is the mean(expected value). of a binomial random variable?

np

What is the standard deviation of a binomial random variable?

sqrt(np(1-p))

What is the 10% condition?

We can use a binomial distribution to model the count of successes in the sample as long as n is less than or equal to 0.1N.

What is the large counts condition?

The probability distribution of a binomial random variable X with n trials and success probability p is normal if:

np>= 10 and n(1-p) >= 10

What is a geometric random variable?

When we perform independent trials of the same random process and record the number of trials needed to get one success.

Binary: The possible outcomes must be success or failure.

Independent: Each trial must be independent.

Same probability: Same P(success) in each trial.

What is the shape of a geometric distribution?

Skewed to the right.

What is the mean and standard deviation of a geometric random variable?

mu(X). =1/p

sigma(X) = sqrt(1-p) / p