This is another proof of Lemma B in Section 7.3 .1 . Let Ui be a random variable with Ui=1 if th

step 1 Okay, so for A, we're going to use the fact that Ui is equal to 1 when i -th population member i

This is another proof of Lemma B in Section 7.3 .1 . Let Ui...

This is another proof of Lemma $\mathrm{B}$ in Section $7.3 .1 .$ Let $U_{i}$ be a random variable with $U_{i}=1$ if the $i$ th population member is in the sample and equal to 0 otherwise. a. Show that the sample mean $\bar{X}=n^{-1} \sum_{i=1}^{N} U_{i} x_{i}$ b. Show that $P\left(U_{i}=1\right)=n / N .$ Find $E\left(U_{i}\right),$ using the fact that $U_{i}$ is a Bernoulli random variable. c. What is the variance of the Bernoulli random variable $U_{i} ?$ d. Noting that $U_{i} U_{j}$ is a Bernoulli random variable, find $E\left(U_{i} U_{j}\right), i \neq j .$ (Be careful to take into account that the sample is drawn without replacement.) e. Find $\operatorname{Cov}\left(U_{i}, U_{j}\right), i \neq j$ f. Using the representation of $\bar{X}$ above, find $\operatorname{Var}(\bar{X})$.

This is another proof of Lemma $\mathrm{B}$ in Section $7.3 .1 .$ Let $U_{i}$ be a random variable with $U_{i}=1$ if the $i$ th population member is in the sample and equal to 0 otherwise.
a. Show that the sample mean $\bar{X}=n^{-1} \sum_{i=1}^{N} U_{i} x_{i}$
b. Show that $P\left(U_{i}=1\right)=n / N .$ Find $E\left(U_{i}\right),$ using the fact that $U_{i}$ is a Bernoulli random variable.
c. What is the variance of the Bernoulli random variable $U_{i} ?$
d. Noting that $U_{i} U_{j}$ is a Bernoulli random variable, find $E\left(U_{i} U_{j}\right), i \neq j .$ (Be careful to take into account that the sample is drawn without replacement.)
e. Find $\operatorname{Cov}\left(U_{i}, U_{j}\right), i \neq j$
f. Using the representation of $\bar{X}$ above, find $\operatorname{Var}(\bar{X})$.

Key Concepts

Sample Mean Representation

The sample mean is a fundamental statistic defined as the sum of observed values divided by the number of observations. In many contexts, especially in survey sampling and inferential statistics, the sample mean is used as an estimator of the population mean. It can also be expressed as a weighted sum of the population values where the weights are indicators for whether a given unit is included in the sample.

Indicator (Bernoulli) Random Variables

Indicator random variables take the value 1 if an event occurs and 0 otherwise. They are the simplest type of Bernoulli random variables, which are used extensively in probability theory and statistics to 'indicate' events. Their binary nature and simple distribution make them a powerful tool for representing selection mechanisms, such as whether a population member is included in a sample.

Bernoulli Random Variables

A Bernoulli random variable models a random process that has exactly two outcomes, typically called 'success' (with probability p) and 'failure' (with probability 1-p). It is characterized by its expectation, p, and its variance, which is given by p(1-p). This concept is fundamental in understanding the behavior of binary outcomes in statistics.

Expectation, Variance, and Covariance

Expectation is the average or mean value of a random variable, while variance measures the spread of the values around this mean. Covariance describes the degree to which two random variables vary together. When analyzing multiple random variables, such as when summing indicator variables, these concepts are essential for determining both individual behavior and the combined variance of the sum.

Sampling Without Replacement

Sampling without replacement means that once a member of the population is selected, it cannot be selected again. This introduces dependence between the selections, affecting probabilities and covariances. In such samples, the joint probability of inclusion for any two distinct members differs from the product of their marginal probabilities because the selection of one unit changes the pool available for the other.

Variance of a Linear Combination of Random Variables

The variance of a linear combination of random variables, such as the sample mean expressed as a weighted sum of values, is computed by summing the variances of the individual terms as well as adding twice the covariances for every pair of distinct terms. This computation is vital in statistics for understanding the precision of estimators and the impact of dependencies among sample elements.

Transcript

Please give Ace some feedback