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# Random variables

AP Stats: UNC (BI), UNC‑3 (EU), VAR (BI), VAR‑5 (EU)

AP Stats: VAR (BI), VAR‑5 (EU), VAR‑5.A (LO), VAR‑5.A.1 (EK), VAR‑5.A.2 (EK), VAR‑5.A.3 (EK), VAR‑5.C (LO), VAR‑5.C.1 (EK), VAR‑5.C.2 (EK), VAR‑5.C.3 (EK), VAR‑5.D (LO), VAR‑5.D.1 (EK)

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Constructing a probability distribution for random variableValid discrete probability distribution examplesProbability with discrete random variable exampleMean (expected value) of a discrete random variableVariance and standard deviation of a discrete random variableMean and standard deviation of a discrete random variable

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Probability with discrete random variablesGet 3 of 4 questions to level up!

Mean (expected value) of a discrete random variableGet 3 of 4 questions to level up!

Standard deviation of a discrete random variableGet 3 of 4 questions to level up!

AP Stats: VAR (BI), VAR‑6 (EU), VAR‑6.A (LO), VAR‑6.A.2 (EK), VAR‑6.A.3 (EK), VAR‑6.B (LO), VAR‑6.B.1 (EK), VAR‑6.B.2 (EK)

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Probabilities from density curves

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Probability in density curvesGet 3 of 4 questions to level up!

Probability in normal density curvesGet 3 of 4 questions to level up!

AP Stats: VAR (BI), VAR‑5 (EU), VAR‑5.F (LO), VAR‑5.F.1 (EK)

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Impact of transforming (scaling and shifting) random variablesExample: Transforming a discrete random variable

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Transforming random variablesGet 3 of 4 questions to level up!

AP Stats: VAR (BI), VAR‑5 (EU), VAR‑5.E (LO), VAR‑5.E.1 (EK), VAR‑5.E.2 (EK), VAR‑5.E.3 (EK)

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Mean of sum and difference of random variablesVariance of sum and difference of random variablesIntuition for why independence matters for variance of sumDeriving the variance of the difference of random variablesCombining random variablesExample: Analyzing distribution of sum of two normally distributed random variablesExample: Analyzing the difference in distributionsCombining normal random variables

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Combining random variablesGet 3 of 4 questions to level up!

Combining normal random variablesGet 3 of 4 questions to level up!

AP Stats: UNC (BI), UNC‑3 (EU), UNC‑3.A (LO), UNC‑3.A.2 (EK), UNC‑3.B (LO), UNC‑3.B.1 (EK)

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Binomial variablesRecognizing binomial variables10% Rule of assuming "independence" between trialsBinomial probability exampleGeneralizing k scores in n attemptsFree throw binomial probability distributionGraphing basketball binomial distributionBinompdf and binomcdf functionsBinomial probability (basic)

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Identifying binomial variablesGet 3 of 4 questions to level up!

Binomial probability formulaGet 3 of 4 questions to level up!

Calculating binomial probabilityGet 3 of 4 questions to level up!

AP Stats: UNC (BI), UNC‑3 (EU), UNC‑3.A (LO), UNC‑3.A.2 (EK), UNC‑3.B (LO), UNC‑3.B.1 (EK), UNC‑3.C (LO), UNC‑3.C.1 (EK)

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Expected value of a binomial variableVariance of a binomial variableFinding the mean and standard deviation of a binomial random variable

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Mean and standard deviation of a binomial random variableGet 3 of 4 questions to level up!

AP Stats: UNC (BI), UNC‑3 (EU), UNC‑3.E (LO), UNC‑3.E.1 (EK), UNC‑3.E.2 (EK), UNC‑3.F (LO), UNC‑3.F.1 (EK)

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Geometric random variables introductionProbability for a geometric random variableCumulative geometric probability (greater than a value)Cumulative geometric probability (less than a value)TI-84 geometpdf and geometcdf functionsProof of expected value of geometric random variable

Practice

Binomial vs. geometric random variablesGet 3 of 4 questions to level up!

Geometric probabilityGet 3 of 4 questions to level up!

Cumulative geometric probabilityGet 3 of 4 questions to level up!

### About this unit

A random variable is some outcome from a chance process, like how many heads will occur in a series of 20 flips (a discrete random variable), or how many seconds it took someone to read this sentence (a continuous random variable). We calculate probabilities of random variables, calculate expected value, and look what happens when we transform and combine random variables.AP® is a registered trademark of the College Board, which has not reviewed this resource.