Discrete Probability Distributions (Cambridge International AS Mathematics)

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)

• $P(X=x)$ 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)

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 $\mathrm{ΣP}(X=x)=1$

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 $\sum P(X=x)=1$

  • Add together all the probabilities and make the sum equal to 1

• To find $P(X=k)$

  • If k is a possible value of the random variable X then $P(X=k)$ will be given in the table
  • If $k$is not a possible value then $P(X=k)=0$

• To find $P(X\leq k)$

  • Identify all possible values, $x_{i}$, that X can take which satisfy $x_{i}\leq k$

  • Add together all their corresponding probabilities

  • $P(X\leq k)=\underset{x_{i}\leq k}{\sum}P(X=x_{i})$

  • Some mathematicians use the notation F(x) to represent the cumulative distribution

    • $F(x)=P(X\leq x)$

• Using a similar method you can find $P(X<k),P(X\geq k)$and $P(X>k)$

• As all the probabilities add up to 1 you can form the following equivalent equations:

  • $P(X<k)+P(X=k)+P(X>k)=1$
  • $P(X>k)=1-P(X\leq k)$
  • $P(X\geq k)=1-P(X<k)$

• To calculate more complicated probabilities such as $P(X^{2}<4)$

  • 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?

• $P(X\leq k)$would be used for phrases such as:

  • At most k, no greater than k, etc

• $P(X<k)$would be used for phrases such as:

  • Fewer than k

• $P(X\geq k)$would be used for phrases such as:

  • At least k , no fewer than k, etc

• $P(X>k)$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

$\mathrm{P} ( X = x ) = \{ \begin{gathered} k x^{2} x = - 3 , - 1 , 2 , 4 \\ 0 otherwise. \end{gathered} )$

(a) Show that $k$ = $\frac{1}{30}$.

(b) Calculate $P(X\leq3).$

(c) Calculate $P(X^{2}<5)$

Answer:

Exam Hint

• 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!