Probability Distributions

Example 1: Creating a Probability Distribution

Create a probability distribution for the sum of two dice rolls.

[4 marks]

Step 1: Create a table of every possible outcome. Note that all of these outcomes are equally likely.

There are thirty-six equally likely outcomes. One of these gives $2$, two of these give $3$, three of these give $4$, four of these give $5$, five of these give $6$, six of these give $7$, five of these give $8$, four of these give $9$, three of these give $10$, two of these give $11$ and one of these gives $12$. So the probability distribution looks like this:

And after simplifying we have our final answer:

Example 2: Probability Function

From the following probability distribution, determine the probability function and the cumulative distribution function.

[4 marks]

The probability function is defined by $f(x)=mathbb{P}(X=x)$, so we need a function that satisfies $f(1)=0.01$, $f(2)=0.03$, $f(3)=0.05$, etc. Note that the function increases by $0.02$ each time, so is linear with a gradient of $0.02$. Hence, we might try $f(x)=0.02x+c$. Subbing in $f(1)=0.01$ gives $c=-0.01$. Hence, the probability function is $f(x)=0.02x-0.01$

To find the cumulative distribution function, we must first add a running total row to our table.

The cumulative distribution function, $F(x)$, must satisfy $F(1)=0.01$, $F(2)=0.04$, $F(3)=0.09$, etc. This is satisfied by $F(x)=dfrac{x^{2}}{100}$.