Probability distribution of a discrete random variable

The probability distribution of a discrete random variable shows all possible values a discrete random variable can have along with their corresponding probabilities. You could use a table to organize the information.

In the lesson about discrete random variable, you conducted a survey asking 200 people about the number of vehicles they own. You came up with the following result.

40 people say that they don't own a car,  100 say they own 1, and 60 say they own 2.

You could start with a table showing the frequency and relative frequency distribution.

  • The relative frequency for owning 0 vehicle is 40/200 = 0.2
  • The relative frequency for owning 1 vehicle is 100/200 = 0.5
  • The relative frequency for owning 2 vehicles is 60/200 = 0.3

Number of vehicles owned Frequency distribution Relative frequency distribution
 0  40  0.2
 1  100  0.5
 2  60  0.3
   N =200  Sum = 1

From the table above, we can pull out the probability distribution. Recall that it will list all the possible values for the random variable and their corresponding probabilities.

Number of vehicles owned or x Probability or P(x)
0 0.2
1 0.5
2 0.3
   \sum_ {}^{}P(X) = 1

Characteristics of the probability distribution of a discrete random variable

You may have noticed the following 2 characteristics after a close examination of the table above.

  • The probability assigned to each value of a discrete random variable is between 0 and 1 inclusively. In other words, 0 ≤ P(x) ≤1
  • The sum of the probabilities is equal to 1

Useful notation

The meaning of P(x = 1) is  probability that a randomly selected person owns 1 car.

P(x = 1) = 0.5

The meaning of P(x > 0) is  probability that a randomly selected person owns at least 1 car.

P(x > 0 ) = P(x = 1) + P(x = 2) = 0.5 + 0.3  = 0.8

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