discrete probability distributions

A Discrete Probability Distribution tells you the various probabilities associated with a discrete random variable. What does this mean? The definition of the word `distribution’ refers to how something is shared out in a group or how it is spread out over an area. Let X be a random variable – a variable whose possible values are the outcomes of a random event (see more on Random Variables). If X is a discrete random variable then its possible values belong to a discrete set of outcomes. If there are three possible outcomes then:

X\in\left\lbrace x_1,x_2,x_3\right\rbrace.

For example, a spinner is numbered 1, 2 and 3 with probabilities of 0.2, 0.35 and 0.45 respectively. This is an example of a discrete random variable and its associated discrete probability distribution.

The Uniform Distribution

The uniform probability distribution describes a discrete distribution where each outcome has an equal probability. Consider a random variable X that has a discrete uniform distribution. X can take one of k values:

X\in\left\lbrace x_1,x_2,x_3,...,x_k\right\rbrace.

If all these values all equally likely then they must each have a probability of 1/k. This is because they must all add up to 1. See Example 1.

The Binomial Distribution

The binomial probability distribution is closely related to Binomial Expansion. It gives the probability distribution of a random variable X that is subject to a number of trials. Each trial is independent (probabilities do not change from trial to trial) has a probability of success or failure. For example, the number of times a head comes up when a coin is tossed repeatedly is a binomial random variable. Each coin toss is a trial and probability of success is the probability of tossing heads.

In general, suppose there are n trials and each trial has a probability of success p. We write X\sim B(n,p) where \sim means ‘distribution’ and ‘B’ means Binomial. The probability of having r successes is P(X=r) and given by:

P(X=r)=\left(\begin{array}{c}n\\r\end{array}\right)p^r(1-p)^{n-r}.

Look up Binomial Expansion to calculate the ‘n choose r’ term in brackets. Now see Example 2. We can use the tables in the Edexcel Formula Booklet (pages 29-33) to find cumulative probabilities. This is where you are looking for the probability of r successes or fewer: P(X\leq r). Look for your value of n down the left hand side of each box in the table. Look for your value of p across the top. You will then see the r (or x) values on the left of each box for which you can read off the corresponding probability inside the table. See Example 3.

Examples

Consider a single dice roll. The set of possible outcomes is {1,2,3,4,5,6} – this is discrete. Each of these outcomes has a probability of 1/6 and therefore represents a discrete uniform distribution.

A coin is tossed 10 times. The number of heads is a binomial random variable. X may take integer between 0 and 10 inclusively. Each of the outcomes has a different probability. To find the probability of tossing 4 heads specifically we put n=10, p=0.5 and r=4 into the formula:

\left(\begin{array}{c}10\\4\end{array}\right)(0.5)^4(0.5)^6=\frac{105}{512}\approx 0.2051.

This is because p and 1-p are both 0.5, i.e. the probability of success (tossing heads is success) and failure (tossing tails is failure) are both the same. The number of trials is n=10 which is the number of tosses. The probability we are looking for is obtaining r=4 successes.

Suppose X\sim B(20,0.25) and we wish to find P(6<X\leq 10). Out of a possible 20 trials, we are looking for the probability of getting 7, 8, 9 OR 10 successes. The Excel Formula booklet with n=20, p=0.25 and x=10 gives the cumulative probability of getting less than or equal to 10 successes as P(X\leq 10)=0.9961. We can also read off the probability of getting less than or equal to 6 successes as P(X\leq 6)=0.7858. The probability of getting 7, 8, 9 or 10 can be found from:

P(6<X\leq 10)=P(X\leq 10)-P(X\leq 6)

Hence,

P(6<X\leq 10)=0.2103.

Videos

https://youtu.be/JrdoWbt4osg

A brief explanation of the uniform distribution and using the Binomial Cumulative Distribution Function on a scientific calculator to find some cumulative binomial probabilities.

https://youtu.be/u885MFUsY6E

Simultaneous Equations where not all the unknowns are found in a Discrete Probability Distributions exam question.

AS Statistics Distributions

A2 Statistics Distributions