The mean of a discrete random variable x is the mean we would expect if the experiment were repeated many times.

The mean value is denoted by μ and obtained with the formula μ = ΣxP (x)

Another term for the mean value of a discrete random variable is expected value.

The expected value is denoted by E (x), i.e. E (x) = ΣxP (x)

In the lesson on the probability distribution of a discrete random variable, we have the following table of probability distribution. Use it to calculate the average number of vehicles owned by people.

Number of vehicles owned or x | Probability or P (x) |

0 | 0.2 |

1 | 0.5 |

2 | 0.3 |

P (x) = 1 |

How to find the mean of the probability distribution of the number of vehicles people own.

x | P (x) | xP (x) = x × P (x) |

0 | 0.2 | 0x0.2 = 0 |

1 | 0.5 | 1 x 0.5 = 0.5 |

2 | 0.3 | 2 x 0.3 = 0.6 |

ΣxP (x) = 0 + 0.5 + 0.6 = 1.1 |

E (x) = 1.1

What does an expected value of 1.1 mean for this situation? That means, on average, you would expect people to own around 1.1 vehicles.

## Another example showing how to find the mean of a discrete random variable

A survey was carried out to find out how often people go to the cinema each week. After interviewing 500 people, the result is shown in the following table. Let x be the number of visits to the cinema per week. If x = 2, the frequency is 75. This means that 75 people went to the cinema twice a week.

x | frequency |

0 | 250 |

1 | 125 |

2 | 75 |

3 | 45 |

4th | 5 |

N = 500 |

P (x = 0) = 250/500 = 0.5

P (x = 1) = 125/500 = 0.25

P (x = 2) = 75/500 = 0.15

p (x = 3) = 45/500 = 0.09

P (x = 4) = 5/500 = 0.01

The following table shows the probability distribution.

x | P (x) |

0 | 0.5 |

1 | 0.25 |

2 | 0.15 |

3 | 0.09 |

4th | 0.01 |

P (x) = 1 |

How to find the mean for the probability distribution of how often people go to the movies.

E (x) = ΣxP (x) = 0 × 0.5 + 1 × 0.25 + 2 × 0.15 + 3 × 0.09 + 4 × 0.01

E (x) = 0 + 0.25 + 0.30 + 0.27 + 0.04

E (x) = 0.86

Based on the expected value of 0.86, the average number of admissions per week is 0.86.