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:

\[ E(X) = \mu_{X} \]

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.

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 \( X \)

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

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: \(V(X) = E(X^2) - \mu_{X}^2 \) |