The expectation of variance derivation

You already know what the random variable is and what the expected value (or expectation) is. If not, please read the background break page.

Expectation rules

The expectation is denoted by the capital letter \( E \).

The expectation of the random variable \( E(X) \) equals the mean of the random variable:

\[ E(X) = \mu_{X} \]
where \( \mu_{X} \) is the mean of the random variable.

Here are some basic expectation rules:

Rule Notes
1 \( E(X) = \mu_{X} = \Sigma xp(x) \) \( p(x) \) is the probability of \( x \) (discrete case)
2 \( E(a) = a \) \( a \) is constant
3 \( E(aX) = aE(X) \) \( a \) is constant
4 \( E(a \pm X) = a \pm E(X) \) \( a \) is constant
5 \( E(a \pm bX) = a \pm bE(X) \) \( a \) and \( b \) are constant
6 \( E(X \pm Y) = E(X) \pm E(Y) \) \( Y \) is another random variable
7 \( E(XY) = E(X)E(Y) \) If \( X \) and \( Y \) are independent

All the rules are quite straightforward and don't need proof.

Expectation of the variance

The expectation of variance is given by:

\[ V(X) = \sigma_{x}^2 = E(X^2) - \mu_{X}^2 \]

where \( V(X) \) is the variance of \( X \)

The proof:

Notes
\( V(X) = \sigma_{X}^2 = E((X - \mu_{X})^2) = \)
\( E(X^2 -2X\mu_{X} + \mu_{X}^2) = \)
\( E(X^2) - E(2X\mu_{X}) + E(\mu_{X}^2) = \) Applied rule number 5: \( E(a \pm bX) = a \pm bE(X) \)
\( E(X^2) - 2\mu_{X}E(X) + E(\mu_{X}^2) = \) Applied rule number 3: \( E(aX) = aE(X) \)
\( E(X^2) - 2\mu_{X}E(X) + \mu_{X}^2 = \) Applied rule number 2: \( E(a) = a \)
\( E(X^2) - 2\mu_{X}\mu_{X} + \mu_{X}^2 = \) Applied rule number 1: \( E(X) = \mu_{X} \)
\( E(X^2) - \mu_{X}^2 \)

The expectation of the body position variance

The body position displacement variance in terms of time and velocity is given by:

\[ V(x) = \Delta t^{2} V(v) \] or \[ \sigma_{x}^2 = \Delta t^{2} \sigma_{v}^2 \]
Where:
\( x \) is the displacement of the body
\( v \) is the velocity of the body
\( \Delta(t) \) is the time interval

The proof:

Notes
\( V(x) = \sigma_{x}^2 = E(x^2) - \mu_{x}^2 = \)
\( E((v\Delta t)^2) - (\mu_{v}\Delta t)^2 = \) Express the body position variance in terms of time and velocity: \( x = \Delta tv \)
\( E(v^{2}\Delta t^{2}) - \mu_{v}^{2}\Delta t^{2} = \)
\( \Delta t^{2}E(v^{2}) - \mu_{v}^{2}\Delta t^{2} = \) Applied rule number 3: \( E(aX) = aE(X) \)
\( \Delta t^{2}(E(v^{2}) - \mu_{v}^{2}) = \)
\( \Delta t^{2}V(v) \) Applied expectation of variance rule: \(V(X) = E(X^2) - \mu_{X}^2 \)
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