You already know what the random variable is and what the expected value (or expectation) is. If not, please read the Essential background I section.
The expectation is denoted by the capital letter \( E \).
The expectation of the random variable \( E(X) \) equals 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.
The expectation of variance is given by:
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 body position displacement variance in terms of time and velocity is given by:
| \( x \) | 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: \(V(X) = E(X^2) - \mu_{X}^2 \) |