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 showing how to find the mean of a discrete random variable

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 in 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.

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