2.8 Constructing Discrete and Cumulative Distributions

Turn Outcomes into Numbers

An outcome need not be numerical: a two-coin result can be HT. A random variable assigns a numerical value to each chance outcome. For two independent fair tosses, define $X$ as the number of heads. Then TT gives 0, HT and TH each give 1, and HH gives 2. These are discrete, countable values. X is not the outcome string; it is the number computed from that string.

The four ordered toss results are equally likely. Combining outcomes that give the same value produces the complete probability distribution:

X: heads	Outcomes	P(X=x)
0	TT	1/4
1	HT, TH	1/2
2	HH	1/4

Every possible value appears, each probability is nonnegative, and the probabilities sum to 1. A probability graph has values 0, 1 and 2 on its horizontal axis and bars of heights 0.25, 0.50 and 0.25. A function may specify those same probabilities at the three values and 0 elsewhere. Tables, graphs and functions are alternative representations of one distribution, not three different models.

Constructing an Empirical Version

The exact table above comes from probability rules. We can also simulate pairs of independent fair tosses and estimate the distribution. Suppose a fresh 40-trial run records X=0 ten times, X=1 eighteen times and X=2 twelve times. Its empirical table is 0.25, 0.45 and 0.30. Dividing each count by 40 makes the probabilities sum to 1. It estimates the exact distribution; differences do not imply that the coin’s model probability changed.

The numerical record and denominator matter. If a run produces no instances of one possible value, include that value with empirical frequency 0 rather than deleting it from the theoretical support. A longer valid run can improve the approximation, but it cannot repair a simulation that fails to use independent fair tosses.

Worked Example: “At Most” Adds Upward

A cumulative distribution records $F(t)=P(X\le t)$. The less-than-or-equal sign includes the value at the boundary. For the exact two-coin model:

Boundary t	Included X values	F(t)
0	0	0.25
1	0, 1	0.75
2	0, 1, 2	1.00

For any real input t, a function gives the whole cumulative picture:

The graph is a staircase: it jumps at the attainable values and is flat between them. Thus $P(X\le1)=0.75$, while $P(X<1)=0.25$. The difference is the probability at X=1. Discrete boundaries matter; a value can have positive probability. The cumulative values never decrease, and the last one is 1 because all possible outcomes have been included.

Building a distribution therefore means defining the variable, finding every possible numerical value, assigning probabilities consistently, and displaying the result with enough labels to know what a value or bar means. A name such as “number of heads” without the probabilities is not yet a distribution.

Try It Yourself

The following original practice applies the ideas above to a fresh setting. Show the reasoning requested in each part. Answers and task-specific rubrics are on the separate solutions page.

Original generated practice for the Fall 2026 course.

Build a Distribution for the Larger Spin

Two fair four-sided spinners, each labeled 1, 2, 3, 4, are spun independently. Let X be the larger of the two numbers. All 16 ordered pairs are equally likely. A separate newly generated simulation produced these 20 pairs:

(4, 2), (2, 1), (1, 4), (2, 4), (3, 4), (2, 4), (2, 3), (2, 3), (2, 1), (4, 4), (3, 4), (3, 3), (1, 1), (4, 3), (2, 1), (1, 3), (3, 3), (2, 4), (4, 4), (3, 4)

Part a

List or systematically organize the 16 outcomes, map them to X, and identify every possible X value. Explain why X is a discrete random variable with numerical chance outcomes.

Your reasoning

Part b

Construct the complete exact probability table and a probability function for X using the equally likely outcome rules. Verify that the probabilities sum to 1.

Your reasoning

Part c

Construct a labeled probability bar graph from your exact table. You may draw it or specify the horizontal-axis values, vertical-axis meaning and every bar height unambiguously.

Your reasoning

Part d

Construct a cumulative table and a cumulative function F(t)=P(X≤t), including ranges below and above the possible values. Calculate P(X≤3) and P(X<3), explaining the boundary difference.

Your reasoning

Part e

Execute the maximum calculation on all 20 simulated pairs. Construct the empirical probability table for every possible X value, state the denominator and compare it with the exact table.

Your reasoning

Worked answer and rubric