Discrete Probability Distributions

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 that a dice lands on when rolled once
  - this has a finite number of outcomes: {1,2,3,4,5,6}

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 discrete probability distribution function or "pdf")
- 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 $\sum P (X = x) = 1$
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 $\frac{1}{n}$

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

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
- $P (X \leq k) = \sum_{x_{i} \leq k} P (X = x_{i})$
- 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 $P (X < k)$ , $P (X > k)$ and $P (X \geq k)$
As all the probabilities add up to 1 you can form the following equivalent equations:
- $P (X < k) + P (X = k) + P (X > k) = 1$
- $P (X > k) = 1 - P (X \leq k)$
- $P (X \geq k) = 1 - P (X < k)$

$P (X \leq k)$ would be used for phrases such as:
- At most , no greater than , etc
$P (X < k)$ would be used for phrases such as:
- Fewer than
$P (X \geq k)$ would be used for phrases such as:
- At least , no fewer than , etc
$P (X > k)$ would be used for phrases such as:
- Greater than , etc

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

$P (X = x) = {\begin{matrix} k x ² \\ 0 \end{matrix} \begin{array}{l} x = - 3, - 1, 2, 4 \\ otherwise . \end{array}$

Show that

k = \frac{1}{30}

4-4-1-ib-ai-aa-sl-discrete-pd-a-we-solution

Calculate

P (X \leq 3)

4-4-1-ib-ai-aa-sl-discrete-pd-b-we-solution

Discrete Probability Distributions (DP IB Maths: AI HL)