Discrete Probability Distributions (CIE A Level Maths: Probability & Statistics 1)

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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)
  • begin mathsize 16px style straight P left parenthesis X equals x right parenthesis end style means "the probability of the random variable X taking the value x"
  • 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)

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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 
    • 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 X  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 begin mathsize 16px style ΣP left parenthesis X equals x right parenthesis equals 1 end style

4-1-1-discrete-probability-distributions-diagram-1

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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 sum straight P left parenthesis X equals x right parenthesis equals 1
    • Add together all the probabilities and make the sum equal to 1
  • To find begin mathsize 16px style P left parenthesis X equals k right parenthesis end style
    • If k is a possible value of the random variable X  then begin mathsize 16px style P left parenthesis X equals k right parenthesis end style will be given in the table 
    • If begin mathsize 16px style k end styleis not a possible value then begin mathsize 16px style P left parenthesis X equals k right parenthesis equals 0 end style
  • To find begin mathsize 16px style P left parenthesis X less or equal than k right parenthesis end style
    • Identify all possible values, begin mathsize 16px style x subscript i end style, that X can take which satisfy begin mathsize 16px style x subscript i less or equal than k end style
    • Add together all their corresponding probabilities
    • straight P left parenthesis X less or equal than k right parenthesis equals sum for x subscript i less or equal than k of straight P left parenthesis X italic equals x subscript i right parenthesis
    • Some mathematicians use the notation F(x) to represent the cumulative distribution
      • begin mathsize 16px style straight F left parenthesis x right parenthesis equals straight P left parenthesis X less or equal than x right parenthesis end style
  • Using a similar method you can find begin mathsize 16px style P left parenthesis X less than k right parenthesis comma space P left parenthesis X greater or equal than k right parenthesis space end styleand begin mathsize 16px style P left parenthesis X greater than k right parenthesis end style
  • As all the probabilities add up to 1 you can form the following equivalent equations:
    • begin mathsize 16px style straight P left parenthesis X less than k right parenthesis plus straight P left parenthesis X equals k right parenthesis plus straight P left parenthesis X greater than k right parenthesis equals 1 end style
  • To calculate more complicated probabilities such as begin mathsize 16px style straight P left parenthesis X squared less than 4 right parenthesis end style 
    • 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?

  • begin mathsize 16px style straight P left parenthesis X less or equal than k right parenthesis end stylewould 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
  • Error converting from MathML to accessible text.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

 straight P left parenthesis X equals x right parenthesis equals open curly brackets table row cell k x squared space space space space space x equals negative 3 comma negative 1 comma 2 comma 4 end cell row cell 0 space space space space space space space space space otherwise. end cell end table close

(a)     Show that  k = 1 over 30.

 

   

(b)      Calculate straight P left parenthesis X less or equal than 3 right parenthesis.

(c)
Calculate straight P left parenthesis X squared less than 5 right parenthesis

4-1-1-discrete-probability-distributions-we-solution-part-1

4-1-1-discrete-probability-distributions-we-solution-part-2

Examiner 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!

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Dan

Author: Dan

Expertise: Maths

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.