## Discrete Random Variables - Statistics 1

Discrete Random Variables are variables that contain discrete data.

Discrete data is when the data can be listed. Discrete data can be counted and is not measured. Examples include:

The number when throwing a six sided die; you can only have the values: 1, 2, 3, 4, 5, 6.

Flipping a coin; you can only have the outcomes: heads or tails.

Discrete Data can be summarised in a Probability Distribution Table.

The probability distribution shows the probability of each value in the list of your discrete data. Probability distribution for throwing a six sided die:

Discrete data is when the data can be listed. Discrete data can be counted and is not measured. Examples include:

The number when throwing a six sided die; you can only have the values: 1, 2, 3, 4, 5, 6.

Flipping a coin; you can only have the outcomes: heads or tails.

Discrete Data can be summarised in a Probability Distribution Table.

The probability distribution shows the probability of each value in the list of your discrete data. Probability distribution for throwing a six sided die:

x |
1 |
2 |
3 |
4 |
5 |
6 |

P(X = x) |
1/6 |
1/6 |
1/6 |
1/6 |
1/6 |
1/6 |

The probabilities in the Probability Distribution table add up to 1. The table contains all the possible values you could have and therefore the probability of all of those values must add up to 1.

E(X) = ΣxP = 1(1/6) + 2(1/6) + 3(1/6) + 4(1/6) + 5(1/6) + 6(1/6) = 3.5

Var(X) = Σx²P – (ΣxP)² = E(X²) – [E(X)]²

Var(X) = 1²(1/6) + 2²(1/6) + 3²(1/6) + 4²(1/6) + 5²(1/6) + 6²(1/6) - 3.5²

Var(X) = 2.92

E(3X + 2) = 3E(X) + 2 = 3(3.5) + 2 = 12.5

Var(4X + 3) = 4²Var(X) = 4²(2.92) = 46.7

__Mean__E(X) = ΣxP = 1(1/6) + 2(1/6) + 3(1/6) + 4(1/6) + 5(1/6) + 6(1/6) = 3.5

__Variance__Var(X) = Σx²P – (ΣxP)² = E(X²) – [E(X)]²

Var(X) = 1²(1/6) + 2²(1/6) + 3²(1/6) + 4²(1/6) + 5²(1/6) + 6²(1/6) - 3.5²

Var(X) = 2.92

__Coding affecting the Mean and Variance__E(3X + 2) = 3E(X) + 2 = 3(3.5) + 2 = 12.5

Var(4X + 3) = 4²Var(X) = 4²(2.92) = 46.7

__Calculating probabilities__x |
1 |
2 |
3 |
4 |
5 |
6 |

P(X = x) |
1/6 |
1/6 |
1/6 |
1/6 |
1/6 |
1/6 |

P(X = 3) = 1/6

P(X ≤ 2) = P(X = 1) + P(X = 2) = 1/6 + 1/6 = 1/3

P(3X – 2 ≤ 7) = P(X ≤ 3) = P(X = 1) + P(X = 2) + P(X = 3) = 1/6 + 1/6 + 1/6 = ½

P(X ≤ 2) = P(X = 1) + P(X = 2) = 1/6 + 1/6 = 1/3

P(3X – 2 ≤ 7) = P(X ≤ 3) = P(X = 1) + P(X = 2) + P(X = 3) = 1/6 + 1/6 + 1/6 = ½

__Cumulative Probability Distribution__x |
1 |
2 |
3 |
4 |
5 |
6 |

P(X = x) |
1/6 |
1/6 |
1/6 |
1/6 |
1/6 |
1/6 |

F(X = x) |
1/6 |
2/6 |
3/6 |
4/6 |
5/6 |
6/6 |

F(3) = P(X = 1) + P(X = 2) + P(X = 3) = 3/6