Suppose you want to estimate $\sigma^2_x$, the variance of a random variable $X$. We want to study the properties of a couple of estimators for the variance. You have an iid sample of $n$ observations. Let $\mu_x=E(X)$. You know that by definition, $\sigma^2_x = E[(X-\mu_x)^2]$. Note that the variance is just an expected value. How do we estimate an expected value? Using a sample mean! So, one estimator for the variance might be

$$ \hat \sigma^2_x = \frac1n \sum_{i=1}^n (X_i - \mu_x)^2 $$

Show that $E[\hat\sigma^2_x]= \sigma^2_x$?

How would I go about solving this?

  • $\begingroup$ Hint: linearity of expectation. $\endgroup$ – Dilip Sarwate Oct 2 '15 at 19:27
  • $\begingroup$ Some resources here and here? $\endgroup$ – SecretAgentMan Oct 10 '18 at 6:15

$$E[\hat \sigma^2_x]=\frac{1}{n} E \left[ \sum_{i=1}^n (X_i^2-2X_i\mu_x+\mu_x^2) \right] = E[X_1^2]-2\mu_x^2+\mu_x^2=\sigma_x^2 $$

  • $\begingroup$ I wonder if the value of $E[(X_i-\mu)^2]$ is called by some special name in the literature. $\endgroup$ – Dilip Sarwate Oct 3 '15 at 2:27

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.