AP Statistics Chapter 16

x1 x2 x3 …… xk p1 p2 p3 pk Discrete Random Variables A discrete random variable X has a countable number of possible values. The probability distribution of X lists the values and their probabilities. Value of x x1 x2 x3 …… xk Probability p1 p2 p3 pk

The probability pi must satisfy the rules of probability.

The mean of a random variable x (x) (also known as expected value) is a weighted average that takes into account that all the outcomes are not equally likely. That does not mean we expect to get that value on any one trial, it means that we expect to get that average value in the long run.

Given a Discrete Probability Distribution: x1 x2 x3 …… xk Value of x x1 x2 x3 …… xk Probability p1 p2 p3 pk

Formulas x = xipi σ2 = (xi - i)2pi

The Standard Deviation (x) of X is the square root of the variance. Both of the previous formulas are on the AP Stat Formula Chart.

To evaluate the formulas we use calculator to help us. 1) Store the sample space in L1 and 2) Store the probabilities in L2. 3) Then: Stat →Calc→1-Var Stats List: L1 FreqList: L2 Calculate.

The variance is the standard deviation squared. The mean is labeled as but is really μx. The standard deviation is σx . The variance is the standard deviation squared.

Rules for Transformations 1) Means: Any equation used to transform data can be used to transform the mean. 2) Variances: Only values that multiply or divide affect variability. Values that add/subtract should be ignored. When working with variances, remember that ALL the values are squares. 3) You should not do math on standard deviations, even if it gives you the correct answer.

Rules for Combining Distributions: 1) Means: You can take the sum or the difference of means, whichever the question calls for. 2) Variances: Variances always add, no matter what because we always create more variability when distributions are combined. (Never take the difference). 3) Standard deviations don’t add. You will get the wrong answer if you add them.

Notes – Chapter 17 Binomial & Geometric Distributions

Discrete Variables Both Binomial and Geometric distributions are discrete. That means the variable takes on a countable set of values without decimals in between.

Binomial Distributions Conditions for a Binomial Distribution: 1) Only two categories (i.e. success or failure) 2) Observations are independent* 3) Probability of success is the same for each observation. 4) Fixed number of n observations

Binomial Distributions *sampling without replacement is okay as long as the population is large

Binomial Distributions If the conditions for a binomial distribution are met, then a binomial distribution can be used. You must show that your conditions are met, in context, before you can use the binomial distribution.

Binomial Distributions Binomial notation is B(n, p) for any binomial distribution, where n is the number of trials and p is the probability

The Binomial Formula You will substitute into the formula when P(x = k) (the first formula that is on the formula chart) but you do not have to hand-calculate.

Binomial Distributions Mean and Standard Deviation of a Binomial Distribution  = np  =

Calculator Steps For P(x = k)

Calculator Steps For Example… In a binomial distribution B(20, .4), P(x = 5) is calculated binompdf(20, .4, 5)

Calculator Steps Any interval: P(x < , >, ≤, ≥)

Calculator Steps For Example, for B(10, .4), P(x < 6) =

Geometric Distributions Conditions 1) Only two categories (i.e. success or failure) 2) Observations are independent* 3) Probability of success is the same for each observation. 4) The variable of interest is the number of trials required to obtain the first success.

Geometric Distributions *sampling without replacement is okay as long as the population is large

Geometric Distributions Rules for calculating Geometric Probabilities: P(X = n) = (1 – p)n-1p P(X > n) = (1 – p)n

Geometric Distributions The mean of a Geometric Random Variable  = 1/p

Geometric Distributions The calculator has a geometpdf that works just like the binomial with arguments p and x. You can also use it with the summation just like binomial.

Binomial & Geometric Distributions Showing your work…. Always check your conditions before using either formula. Be specific & use context. 2) Show as much work as possible. If you are using the calculator, show your intervals or inequalities.