Edexcel A Level Maths: Statistics:复习笔记4.1.1 Discrete Probability Distributions

Discrete Random Variables

What is a discrete random variable?

  • A random variable is a variable whose value depends on the outcome of a random event
    • The value of the random variable is not known until the event is carried out (this is what is meant by 'random' in this case)
  • Random variables are denoted using upper case letters (X , Y , etc )
  • Particular outcomes of the event are denoted using lower case letters ( x, y, etc)
  • means "the probability of the random variable X taking the value "
  • A discrete random variable (often abbreviated to DRV) can only take certain values within a set
    • Discrete random variables usually count something
    • Discrete random variables usually can only take a finite number of values but it is possible that it can take an infinite number of values (see the examples below)
  • Examples of discrete random variables include:
    • The number of times a coin lands on heads when flipped 20 times
      (this has a finite number of outcomes: 0,1,2,…,20)
    • The number of emails a manager receives within an hour
      (this has an infinite number of outcomes: 1,2,3,…)
    • The number of times a dice is rolled until it lands on a 6
      (this has an infinite number of outcomes: 1,2,3,…)
    • The number on a bingo ball when one is drawn at random
      (this has a finite number of outcomes: 1,2,3…,90)

Probability Distributions (Discrete)

What is a probability distribution?

  • A discrete probability distribution fully describes all the values that a discrete random variable can take along with their associated probabilities
    • This can be given in a table (similar to GCSE)
    • Or it can be given as a function (called a probability mass function)
    • They can be represented by vertical line graphs (the possible values for   along the horizontal axis and the probability on the vertical axis)
  • The sum of the probabilities of all the values of a discrete random variable is 1
    • This is usually written
  • A discrete uniform distribution is one where the random variable takes a finite number of values each with an equal probability
    • If there are n values then the probability of each one is

Cumulative Probabilities (Discrete)

How do I calculate probabilities using a discrete probability distribution?

  • First draw a table to represent the probability distribution
    • If it is given as a function then find each probability
    • If any probabilities are unknown then use algebra to represent them
  • Form an equation using
    • Add together all the probabilities and make the sum equal to 1
  • To find
    • If is a possible value of the random variable X then  will be given in the table
    • If is not a possible value then
  • To find
    • Identify all possible values, , that X can take which satisfy
    • Add together all their corresponding probabilities
    • Some mathematicians use the notation F(x) to represent the cumulative distribution
  • Using a similar method you can find
  • As all the probabilities add up to 1 you can form the following equivalent equations:
  • To calculate more complicated probabilities such as
    • Identify which values of the random variable satisfy the inequality or event in the brackets
    • Add together the corresponding probabilities

How do I know which inequality to use?

  • would be used for phrases such as:
    • At most k, no greater than k, etc
  • would be used for phrases such as:
    • Fewer than k
  • would be used for phrases such as:
    • At least k  , no fewer than k, etc
  • would be used for phrases such as:
    • Greater than k, etc

Worked Example

The probability distribution of the discrete random variable  is given by the function

(a)     Show that 

(b)   Calculate

(c)   Calculate


Exam Tip

  • Try to draw a table if there are a finite number of values that the discrete random variable can take
  • When finding a probability, it will sometimes be quicker to subtract the probabilities of the unwanted values from 1 rather than adding together the probabilities of the wanted values
  • Always make sure that the probabilities are between 0 and 1, and that they add up to 1!