Unit 5 of 13

Expectation, Spread, Sums and Averages

  • 6 lessons
  • Videos 31 to 36

Expectation is linear even for dependent variables. This unit uses indicators to find expected counts, defines variance and standard deviation, and builds the distributions of sums and sample means.

Linearity gives the mean of a sum without any independence assumption, and indicator variables turn expected counts into sums of probabilities. Variance averages squared deviations from the mean, and the standard deviation returns to the original units, with simple rules for shifts and changes of scale. The distribution of a sum is built from ordered pairs, and the variances of independent variables add. The unit ends with the sampling distribution of a sample mean, whose variance is the population variance divided by the sample size for independent and identically distributed observations.

Every lesson will be one video. The videos are coming soon, and each lesson page already has the definitions and an example for every key term.

Linearity and Indicators

  1. Linearity of Expectation

    Why the expected value of a sum or linear combination is the same combination of expected values when each is finite, with no independence needed, and why products and squares need their own calculation.

    Video 31

  2. Indicator Random Variables

    How an indicator turns an event into a zero-one random variable whose expectation is its probability, and how combining indicators gives expected counts, the indicator of a union and expected numbers of distinct categories.

    Video 32

Variance and Standard Deviation

  1. Variance

    How variance measures spread as the expected squared deviation from the mean, how the second moment gives the computational formula, why variance has squared units, and why a finite variance needs a finite second moment.

    Video 33

  2. Standard Deviation and Changing Scale

    How the standard deviation returns spread to the original units, how it differs from the mean absolute deviation, and how shifts, changes of scale and affine transformations change the mean, variance and standard deviation.

    Video 34

Sums and Sample Means

  1. Sums of Discrete Random Variables

    How to find the distribution of a sum of two discrete random variables with an ordered-pair table and discrete convolution, why variances add for independent variables, and how a cross term enters the variance of a dependent sum.

    Video 35

  2. Sample Means and Sampling Distributions

    What a random sample of independent and identically distributed observations is, how a statistic such as the sample mean has its own sampling distribution, and how the standard error of the sample mean depends on the sample size.

    Video 36