The expected value of the square of the sample mean?

[pandaBee]

Homework Statement

In my notes I keep stumbling upon this equation:

Equation 1: E(X-bar^2) = (σ^2)/n + μ^2

I was wondering why the above equation is true and how it is derived.

The Attempt at a Solution

E(X-bar^2)

##Summations are from i/j=1 to n

= E[(Σx_i/n)^2)]

= E[(Σx_i/n)(Σx_j/n)]

=(1/n^2)E[(Σx_i)(Σx_j)]

=(1/n^2)ΣΣE(x_i*x_j)

=E(x_1^2) + E( x_2^2 + ... + E(x_n^2)
+ E(x_1*x_2) + E(x_1*x_3) + ... : Equation 2

I'm stuck at this point. Could someone show me how you get to Equation 1 from Equation 2? Assuming that my derivation to Equation 2 is correct.

 

[Ray Vickson]

Homework Statement

In my notes I keep stumbling upon this equation:

Equation 1: E(X-bar^2) = (σ^2)/n + μ

I was wondering why the above equation is true and how it is derived.

The Attempt at a Solution

E(X-bar^2)

##Summations are from i/j=1 to n

= E[(Σx_i/n)^2)]

= E[(Σx_i/n)(Σx_j/n)]

=(1/n^2)E[(Σx_i)(Σx_j)]

=(1/n^2)ΣΣE(x_i*x_j)

=E(x_1^2) + E( x_2^2 + ... + E(x_n^2)
+ E(x_1*x_2) + E(x_1*x_3) + ... : Equation 2

I'm stuck at this point. Could someone show me how you get to Equation 1 from Equation 2? Assuming that my derivation to Equation 2 is correct.

If you mean
$$E \overline{X}^2 = \frac{\sigma^2}{n} + \mu$$
then that cannot possibly be right! For one thing, the "dimensions" don't match. For example, if $X$ is measured in meters, then $\sigma^2$ is measured in $\text{meters}^2$, while $\mu$ is in meters.

The easiest approach is to recognize that the mean square and variance of any random variable $Y$ are connected by the equation
$$\text{Var}(Y) = E(Y^2) - (E Y)^2, \; \text{or} \; E(Y^2) = \text{Var}(Y) + (EY)^2$$

Apply this to $Y = \bar{X} = \sum X_i / n$. Assuming the $X_i$ are independent and identically distributed there are easy and well-known formulas for $EY$ and $\text{Var} (Y)$ in terms of $\mu$, $\sigma$ and $n$. That will tell you the value of $E \overline{X}^2$.

On the other hand, if you meant that you want to know
$$E \overline{X^2} = E \sum X_i^2 / n,$$
then you have a different set of formulas to use. However, you can approach it again using the relationship between mean square and variance.

 

[SteamKing]

Homework Statement

In my notes I keep stumbling upon this equation:

Equation 1: E(X-bar^2) = (σ^2)/n + μ

I was wondering why the above equation is true and how it is derived.

I think your Equation 1 may contain a slight error, as Ray pointed out

I think the correct equation is E(x-bar)² = (σ² / n) + μ²

 

[pandaBee]

Yes, you are both correct, the correct equation was actually

Equation 1: E(X-bar^2) = (σ^2)/n + μ^2
The mu had to be squared.

In fact, I can see that I have made a huge blunder in how I was looking at this problem. Thank you both for the insight!