# Mean of a discrete random variable

The mean of a discrete random variable x is the average value that we would expect to get if the experiment is repeated a large number of times.

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

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

The expected value is denoted by E(x), so E(x) = ΣxP(x)

In the lesson about probability distribution of a discrete random variable, we have the probability distribution table below. Use it to compute the mean 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 |

ΣP(x) = 1 |

Here is how to calculate the mean for the probability distribution of number of vehicles owned by people.

x | P(x) | xP(x) = x × P(x) |

0 | 0.2 | 0 × 0.2 = 0 |

1 | 0.5 | 1 × 0.5 = 0.5 |

2 | 0.3 | 2 × 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? It means that on average, you would expect people to own about 1.1 vehicles.

**Another example**

A survey was conducted to find out how many times people go to the movie theater per week. After interviewing 500 people, the result is shown is the table below. Let x be the number of times people go to the movie theater per week. When x = 2, the frequency is 75. This means that 75 people went to the movie theater twice per week.

x | Frequency |

0 | 250 |

1 | 125 |

2 | 75 |

3 | 45 |

4 | 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 table below shows the probability distribution.

x | P(x) |

0 | 0.5 |

1 | 0.25 |

2 | 0.15 |

3 | 0.09 |

4 | 0.01 |

ΣP(x) = 1 |

Here is how to calculate the mean for the probability distribution of number of times people go to the movie theater.

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 mean number of times people will go to the movie theater per week is 0.86.