AP Statistics Chapter 6

Probability Model

describes the possible outcomes of a chance process and the likelihood that those outcomes will occur

• usually in table and the table has the probability

Random Variable

takes numerical values that describ ethe outcomes of some chance process

Probability Distribution

means the the possible values and their probabilities of a random variable

2 types of random variables

discrete and continuous

Discrete Random Variable

takes a fixed set of possible values with gaps between

• countable 

• Ex. number of sixes on a die 

• Ex. number of children in a family —> cant be ½ children 

Probability Requirements

• has to be between 0 and 1 

• sum of the probability are 1

When analyzing discrete random variables

describe the shape, center, and spread—> any outliers

Mean of a Discrete Random Variable

Multiply each value by its probability and then add it up

Standard Deviation of a Discrete Random Variable

Square root all over the sum of (Value - Mean) times Probability of Value

Variance

Standard Deviation squared

Continuous Random Variables

Takes on all values in an interval of numbers

• all values in an interval 

• measurable 

• only intervals have probability 

Interpretation for Mean Variables

If many many ____ are randomly selected, we expect the average amount of _____ to be about ___. 

Interpretation for Standard Deviation Variables

If many many ____ are randomly selected, the number of ____ will typically vary from the mean by about ___. 

Transforming: Multiplying a Random Variable by a Constant

• multiplies/divides measures of center and location (mean, median, quartiles)

• multiplies/divides measures of spread (range, IQR, standard deviation) 

• does not change the shape of the distribution

• multiply by b

Transforming: Adding a Constant to a Random Variable

• adds to measures of center and location (mean, median)

• does not change measures of spread (range, IQR, standard dev.) 

• does not change shape 

• add by a 

Linear Transformations

y = a + b(mu)x —> linear transformation of the random variable X

• the probability distribution of Y has the same shape as the probability distribution of X 

The only way to determine the probability for any value if..

X and Y are independent random variables

How to know if X and Y are independent random variables

If knowing that event X has occurred and doesn’t tell us anything about the occurrence of any event involving Y alone, then they are independent 

Mean of the Sum of Random Variables

Mean of sum = the sum of the means

Variance of the Sum of Random Variables

Variance of the sum = sum of their variances

When can you add standard deviations

NEVER.

Mean of the Difference of Random Variables

Mean of difference = difference of their means (subtract the means)

Variance of the Difference of Random Variables

Variance of the Difference = Sum of their variances (add)

• to find the standard deviation—> square root the result