The problem involves showing that $E(S) \leq \sigma$, and that if $\sigma^2 > 0$, then $E(S) < \sigma$.

We are given the following:

• A random sample $X_1, X_2, \dots, X_n$ from a population with variance $\sigma^2$.
• The sample mean is $\bar{X}$, and the sample variance is $S^2$, which is defined as:

$S^2 = \frac{1}{n-1} \sum_{i=1}^n (X_i - \bar{X})^2.$

We also know that:

$E(S^2) = \sigma^2.$

Part 1: Prove $E(S) \leq \sigma$

To prove this, we will use Jensen's inequality. Jensen's inequality states that for a convex function $g(x)$, the following holds:

$E(g(X)) \geq g(E(X)).$

In this case, $S = \sqrt{S^2}$, and the square root function is concave, meaning $g(x) = \sqrt{x}$ satisfies the reverse inequality:

$E(\sqrt{S^2}) \leq \sqrt{E(S^2)}.$

Using this, we get:

$E(S) = E(\sqrt{S^2}) \leq \sqrt{E(S^2)} = \sqrt{\sigma^2} = \sigma.$

Thus, we have shown that $E(S) \leq \sigma$.

Part 2: Prove that if $\sigma^2 > 0$, then $E(S) < \sigma$

To prove strict inequality, note that $S$ is a random variable, while $\sigma$ is a constant. $S^2$ is an unbiased estimator of $\sigma^2$, meaning $E(S^2) = \sigma^2$. However, because $S$ is random, it will take values both below and above $\sigma$, but since $S^2$ has a nonzero variance when $\sigma^2 > 0$, we will have:

$E(S) < \sigma.$

This is because the concavity of the square root function ensures that the expected value of the square root is strictly less than the square root of the expected value unless $S^2$ is a degenerate distribution (i.e., with zero variance, which would only happen if $\sigma^2 = 0$).

Thus, for $\sigma^2 > 0$, we have $E(S) < \sigma$.

Would you like more details on Jensen's inequality or any further explanation on how the sample variance behaves?

5 Related Questions:

1. What is Jensen's inequality, and why does it apply in this context?
2. How is the sample variance $S^2$ an unbiased estimator of the population variance?
3. Why is the square root function concave, and how does this affect the expectation?
4. What happens to the relationship between $E(S)$ and $\sigma$ as the sample size $n$ increases?
5. How does the strict inequality $E(S) < \sigma$ change if $\sigma^2 = 0$?

Tip: Remember that Jensen's inequality is a powerful tool whenever you are dealing with expectations of convex or concave functions.