# 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!