AP StatisticsAP Statistics

Chapter 7 NotesChapter 7 Notes

Random VariablesRandom Variables

Random VariableRandom Variable– A variable whose value is a numerical A variable whose value is a numerical

outcome of a random phenomenon.outcome of a random phenomenon.

Discrete Random VariableDiscrete Random Variable– Has a countable number of outcomesHas a countable number of outcomes– e.g. Number of boys in a family with 3 childrene.g. Number of boys in a family with 3 children

(0, 1, 2, or 3)(0, 1, 2, or 3)

Probability DistributionProbability Distribution

Lists the values of a discrete random Lists the values of a discrete random variable and their probabilities.variable and their probabilities.

Value of X: Value of X: xx11 xx22 xx33 xx4 4 . . . .. . . . xxkk

P(X) :P(X) : pp11 pp22 pp33 pp44. . . .. . . . ppkk

Example of a Probability Example of a Probability Distribution (Discrete RV)Distribution (Discrete RV)

XXage when male college students age when male college students began to shave regularly.began to shave regularly.

XX 11 12 13 14 15 16 17 18 19 11 12 13 14 15 16 17 18 19 2020 p(x).013 0 .027 .067 .213 .267 .240 .093 .067 .013p(x).013 0 .027 .067 .213 .267 .240 .093 .067 .013

Continuous Random VariableContinuous Random Variable

Takes on all values in an interval of Takes on all values in an interval of numbers.numbers.– e.g. women’s heightse.g. women’s heights– e.g. arm lengthe.g. arm length

Probability Distribution for Continuous RVProbability Distribution for Continuous RV– Described by a density curve.Described by a density curve.– The probability of an event is the area under a The probability of an event is the area under a

density curve for a given interval.density curve for a given interval.– e.g. a Normal Distributione.g. a Normal Distribution

MeanMean

The mean of a random variable is The mean of a random variable is represented by represented by μμxx, , μμyy, etc., etc.

The mean of X is often called the The mean of X is often called the expected expected value value of X.of X.– The “expected value” does not have to be a The “expected value” does not have to be a

number that can possibly be obtained, number that can possibly be obtained, therefore you can’t necessarily “expect” it to therefore you can’t necessarily “expect” it to occur. occur.

Mean FormulaMean Formula

For a discrete random variable with the For a discrete random variable with the distribution.distribution.

μμxx = ∑ x = ∑ xi i ppii

X:X: x1x1 x2x2 x3 x3 x4 . . . . xk x4 . . . . xk P(X):P(X): p1p1 p2 p2 p3 p3 p4. . . . pk p4. . . . pk

Example of a Probability Example of a Probability Distribution (Discrete RV)Distribution (Discrete RV)

XXage when male college students age when male college students began to shave regularly.began to shave regularly.

XX 11 12 13 14 15 16 17 18 19 11 12 13 14 15 16 17 18 19 2020 p(x).013 0 .027 .067 .213 .267 .240 .093 .067 .013p(x).013 0 .027 .067 .213 .267 .240 .093 .067 .013

Variance/ Standard DeviationVariance/ Standard Deviation

The variance of a random variable is The variance of a random variable is represented by represented by σσ22

xx and the standard and the standard

deviation by deviation by σσxx..

For a discrete random variable…For a discrete random variable…

σσ22xx = ∑(x = ∑(xii – – μμxx))2 2 ppii

Law of Large NumbersLaw of Large Numbers

As the sample size increases, the sample As the sample size increases, the sample mean approaches the population mean.mean approaches the population mean.

Rules for means of Random Rules for means of Random VariablesVariables

1.1.μμa+bxa+bx = a + b = a + bμμxx– If you perform a linear transformation on every If you perform a linear transformation on every

data point, the mean will change according to data point, the mean will change according to the same formula.the same formula.

2. 2. μμX ± YX ± Y = = μμXX ± ± μμYY– If you combine two variables into one If you combine two variables into one

distribution by adding or subtracting, the mean distribution by adding or subtracting, the mean of the new distribution can be calculated using of the new distribution can be calculated using the same operation.the same operation.

Rules for variances of Random Rules for variances of Random Variables Variables

1. 1. σσ22a + bxa + bx = b = b22σσ22

xx

2. 2. σσ22X + YX + Y = = σσ22

X X + + σσ22Y Y

σσ22X - YX - Y = = σσ22

X X + + σσ22YY

– X and Y must be independentX and Y must be independent

Any linear combination of independent Any linear combination of independent Normal random variables is also Normally Normal random variables is also Normally distributed.distributed.