Katkat
Back

Derive mean and variance of x̄

LinkedInData Scientist · Technical Screen
MediumData ScientistTechnical ScreenStatistics & Math

Last updated

This question tests your ability to derive the expected value and variance of the sample mean for a sequence of random variables, accounting for the effect of correlation between them.

Derivation of Sample Mean Properties

Suppose you have a collection of $n$ random variables $X_1, X_2, ..., X_n$. Each variable has a mean $E[X_i] = \mu$ and a variance $Var(X_i) = \sigma^2$ for all $i = 1, ..., n$.

Let the sample mean be defined as: x_bar = (1/n) * sum(X_i for i in 1 to n)

Part 1: Expected Value

Derive $E[\bar{x}]$ in terms of $\mu$.

Part 2: Variance

Assume that each pair of distinct variables $(X_i, X_j)$ where $i \neq j$ has a constant covariance $Cov(X_i, X_j) = \rho\sigma^2$, where $\rho$ is the correlation coefficient between any two distinct observations.

Derive $Var(\bar{x})$ in terms of $n, \sigma^2,$ and $\rho$.

Part 3: Interpretation

Explain what happens to $Var(\bar{x})$ as $n \to \infty$ under two scenarios:

  1. When observations are independent ($\rho = 0$).
  2. When observations are perfectly positively correlated ($\rho = 1$).

Hints

Hint 1

Use the linearity of expectation property for the first part: $E[aX + bY] = aE[X] + bE[Y]$.

Hint 2

Recall that the variance of a sum is the sum of the variances plus the sum of all pairwise covariances: $Var(\sum X_i) = \sum Var(X_i) + \sum_{i \neq j} Cov(X_i, X_j)$.

Hint 3

Remember that there are $n(n-1)$ pairs of $(X_i, X_j)$ where $i \neq j$ in the expansion of the variance of the sum.

Follow-up questions

  • How would your derivation change if the variables were not identically distributed (i.e., they had different means $\mu_i$)?
  • In a real-world dataset, why might observations be correlated (violating the i.i.d. assumption)?
  • If you are collecting data through a survey and the responses are correlated due to clustering (e.g., households), how does this affect the 'Effective Sample Size'?