Probability distributions

​​In a nutshell

Probability distributions are often used to describe the different probabilities for a certain event. The discrete uniform distribution is a specific example of a probability distribution that has many applications.

Definitions

Here are a list of definitions you should know relating to probability distributions.

Representing probability distributions

The two main ways to represent a discrete probability distribution are a probability mass function and a table.

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Example 1

The result of a fair six-sided die is represented with a random variable, $$X$$. Represent the distribution of $$X$$.

First, identify the outcomes and probabilities.

A fair die will land on the numbers $$1,2,3,4,5,6$$ with an equal probability of $$\dfrac{1}{6}$$.

Represent the probabilities using a function:

​$$P(X=x)=\dfrac{1}{6},\space x\in \{1,2,3,4,5,6\}$$​​

As a table:

Note: The notation $$P(X=x)$$ means "the probability that the random variable $$X$$ takes the specific outcome $$x$$". It is also correct to write $$x=1,2,3,4,5,6$$ instead of $$x=\{1,2,3,4,5,6\}$$.

The discrete uniform distribution

The discrete uniform distribution is a specific distribution that only takes certain values and every single probability is the same. The random variable $$X$$ in the example above is an example of a discrete uniform distribution.

Probability distribution problems

Sum of probabilities

When solving problems involving a probability distribution, you may have to use the fact that probabilities always add up to $$1$$. With the notation used, this can be written as:

$$\boxed{\sum_{x}{P(X=x)}=1}$$​​

Example 2

The probability mass function representing a random variable $$X$$ is given to be:

​$$P(X=x)=kx,\space x\in \{1,2,3\}$$​

Find the value of $$k$$.

First, find out all of the probabilities:

​$$P(X=1)=k(1)=k\\P(X=2)=k(2)=2k\\P(X=3)=k(3)=3k$$​​

Then, use the fact that the probabilities add up to $$1$$ to form an equation in $$k$$:

​$$k+2k+3k=1\\6k=1\\\underline{k=\dfrac{1}{6}}$$​​

Inequalities

You may be asked to find the probability that a random variable satisfies a given inequality. To approach these questions, think about which specific outcomes satisfy the inequality and add them up.

Example 3

Using the probability distribution in the above example, find $$P(0.5\le X<3)$$.

Find out which outcomes satisfy the inequality.

The only values that satisfy $$0.5\le X < 3$$ are $$X=1$$ and $$X=2$$.

Add up the probabilities:

​$$\begin{aligned}P(0.5\le X< 3)&=P(X=1)+P(X=2)\\&=k+2k\\&=3k\\&=3(\dfrac{1}{6})\\&=\dfrac{1}{2}\end{aligned}$$​​

​$$\underline{P(0.5\le X < 3) = \dfrac{1}{2}}$$​​