Discrete Probability Distributions (DP IB Analysis & Approaches (AA): HL): Revision Note

Discrete probability distributions

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 occurs (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)

• $\mathrm{P}(X=x)$ 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 often count something

  • A 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: {0,1,2,3,…}

  • 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}

What is a probability distribution of a discrete random variable?

• 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")

  • Or it can be represented by vertical line graph (the possible values 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 $\sum \mathrm{P}(X=x)=1$

• 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 $\frac{1}{n}$

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 $\sum \mathrm{P}(X=x)=1$

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

• To find $\mathrm{P}(X=k)$

  • If $k$ is a possible value of the random variable $X$ then $\mathrm{P}(X=k)$ will be given in the table

  • If $k$ is not a possible value then $\mathrm{P}(X=k)=0$

• To find $\mathrm{P}(X\le k)$

  • Identify all possible values, $x_i$, that $X$ can take which satisfy $x_i\le k$

    • Note that this includes $k$

  • Add together all their corresponding probabilities

  • $\mathrm{P}(X\le k)=\sum _{x_i\le k}\mathrm{P}(X=x_i)$

  • Some mathematicians use the notation $\mathrm{F}\left(x\right)$ to represent the cumulative distribution

    • $\mathrm{F}\left(x\right)=\mathrm{P}(X\le x)$

• Using a similar method you can find $\mathrm{P}(X<k)$, $\mathrm{P}(X>k)$ and $\mathrm{P}(X\ge k)$

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

  • $\mathrm{P}(X<k)+\mathrm{P}(X=k)+\mathrm{P}(X>k)=1$

  • $\mathrm{P}(X>k)=1-\mathrm{P}(X\le k)$

  • $\mathrm{P}(X\ge k)=1-\mathrm{P}(X<k)$

How do I know which inequality to use? 

• $\mathrm{P}(X\le k)$ would be used for phrases such as:

  • At most , no greater than , etc

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

  • Fewer than, etc.

• $\mathrm{P}(X\ge k)$ would be used for phrases such as:

  • At least , no fewer than , etc

• $\mathrm{P}(X>k)$ would be used for phrases such as:

  • Greater than , etc