Empirical Discrete Probability Distributions | Free Discrete Mathematics Lesson (4 College Credits) | UPI Study

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Empirical Discrete Probability Distributions

Discrete Mathematics · Chapter 6: Random Variables and Expected ValueSection 6.4 · lesson 38 of 878 min · free with a free account

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In this lesson

Distributions built from what actually happened.

  • Compute relative frequencies from observed counts
  • Build and verify an empirical distribution
  • Compare empirical results with a theoretical model
  • Allow for sampling variability before drawing conclusions

Transcript

Everything said in the video, section by section (1,072 words). Read along, or search it.

Random Variables and Expected Value

Empirical Discrete Probability Distributions

Section six point three built distributions from assumptions. This section builds them from data. We'll turn raw observed counts into a distribution using relative frequency, and then compare what actually happened with what a theoretical model predicted. And we'll see why some gap between the two is expected, even when the model is right.

The Opening Problem

The Question 6.3 Left Open

Section six point three left an open question. A game studio's loot table was verified mathematically. Its probabilities summed to one, built from a stated assumption. But after the feature shipped, analysts logging real player outcomes noticed tier six being awarded less often than the model predicted. Resolving that gap needs a second kind of distribution altogether. One built not from an assumption about how the mechanic is supposed to behave, but from what it actually did, across real logged outcomes. That's an empirical distribution. And the tool connecting raw counts to a distribution table is relative frequency.

Definition 6.4.1

Relative Frequency

Suppose a random variable X is observed over n independent trials, and a value x occurs in f of x of them. That count is called the frequency of x. The relative frequency of x is its frequency divided by n. An empirical distribution lists, for every observed value, its relative frequency, in place of a theoretically derived probability. The table takes exactly the same form as a P M F. And a correctly built one satisfies the same two conditions. Every relative frequency is non-negative. And since each trial produces exactly one recorded value, they sum to exactly one.

Worked Example 6.4.1

200 Logged Loot-Box Openings

Here's a worked data set. A studio logs which of six loot tiers was awarded, across two hundred real openings. Tier one came up forty-four times, tier two forty-two, tier three forty, tier four thirty-two, tier five thirty, and tier six just twelve. As a first check, those counts should account for all two hundred openings. They do. Dividing each frequency by two hundred gives the empirical distribution. Zero point two two, zero point two one, zero point two, zero point one six, zero point one five, and zero point zero six. And since the frequencies sum to two hundred, the relative frequencies sum to exactly one. The table is valid, before it's compared with anything else.

Worked Example 6.4.2

Support Ticket Resolution

A second example. A helpdesk logs how many email exchanges each of one hundred fifty resolved tickets needed. Sixty-two needed one exchange, forty-five needed two, twenty-eight needed three, and fifteen needed four. That's all one hundred fifty. The relative frequencies are about zero point four one three, exactly zero point three, about zero point one eight seven, and exactly zero point one. To verify, sum the exact fractions before rounding. One hundred fifty over one hundred fifty is exactly one, even though several entries only round to three decimal places.

Table 6.4.1

Model Versus Data, Tier by Tier

This table places the loot data directly alongside the theoretical model from section six point three. The model gives tiers one to three zero point two each, tiers four and five zero point one five, and tier six zero point one. Tier one came in zero point zero two above its model value. Tiers two and four, zero point zero one above. Tiers three and five match exactly. And tier six came in zero point zero four below. It was predicted at zero point one, and observed at zero point zero six.

Figure 6.4.1

Tier 6 Falls Short

Here's the same comparison as a picture. The green bars are the theoretical model, and the gold bars are the empirical results from two hundred openings. For tiers one through five, the pairs track each other closely. Tier six stands out. Its gold bar falls noticeably short of the predicted zero point one.

Resolving 6.3's Question

The Arithmetic Was Never Wrong

So does that gap mean the theoretical table's arithmetic was wrong? No. It still sums to one, exactly as verified. That internal check was never in question. What the comparison reveals is a separate, empirical question. Does the assumption behind the model hold? Does the shipped mechanic really behave like the intended weighted model? Tiers three and five match their theoretical values exactly, and tiers one, two, and four land within zero point zero two. But tier six's empirical rate, zero point zero six, falls well below its theoretical zero point one. That's a gap worth investigating. It could be a bug in the reward code, or a rate miscalibrated during implementation. Or it could simply be chance.

Sampling Variability

An Estimate, Not a Guarantee

That last possibility matters. Some divergence is expected, even when a theoretical model is exactly correct. With only two hundred trials, relative frequency is an estimate, not a guarantee. And estimates carry sampling variability, which shrinks as more data is collected. So a single modest gap doesn't, by itself, prove the model wrong. A persistent gap, one that fails to shrink as more openings are logged, is much stronger evidence that the underlying assumption needs revisiting.

6.4.4 Worked Comparison

A Staffing Assumption Meets the Data

Here's one more comparison. Before collecting any data, a helpdesk manager assumed, for staffing, that a ticket would need one, two, three, or four exchanges with equal likelihood. That's zero point two five each. Compare that with the one hundred fifty real tickets. The equal-likelihood guess badly underestimates single-exchange tickets. It assumed zero point two five, and the actual rate is zero point four one three, nearly double. And it badly overestimates four-exchange tickets. It assumed zero point two five, and the actual rate is zero point one, less than half. Staffing built on the guess would have overallocated to long tickets that are comparatively rare, while underestimating the quick, single-exchange work. The empirical distribution, built from what tickets actually required, is a far better foundation for future staffing.

Takeaways

Count, Divide, Compare

To summarize. An empirical distribution is built from observed data. Each value's relative frequency estimates its probability, in the same table form as a P M F. A correctly built one meets the same two conditions. Every entry is non-negative, and they sum to exactly one. Comparing it with a theoretical model tests whether the model's assumptions match reality. That's a separate question from whether the model is internally valid. And some discrepancy is expected even when a model is right, especially with small samples. Here's something to think about. Suppose an analyst reruns the loot comparison using only the first twenty of the two hundred openings. Would you expect the relative frequencies to match the model more closely, less closely, or about the same? And what does that suggest about how much data is enough, before treating a discrepancy as meaningful, rather than coincidental?

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FAQ

Questions students ask us

About Empirical Discrete Probability Distributions

What is an empirical probability distribution?
An empirical distribution is a table that lists each observed value of a random variable with its relative frequency, instead of a theoretically derived probability. It has the same form as a PMF. Every entry is non-negative, and the entries sum to 1, since each trial produces exactly one recorded value.
How do you calculate relative frequencies from observed counts?
Divide each value's frequency by the total number of trials: \text{relative frequency} = f(x)/n. For example, if tier 6 appeared 12 times in 200 loot-box openings, its relative frequency is 12/200 = 0.06. Check that the counts add up to n, so the relative frequencies sum to 1.
How do I check that an empirical distribution is valid if the decimals are rounded?
Check that every relative frequency is non-negative and that the exact fractions sum to 1. Do this before rounding. In the helpdesk data, 62/150 is about 0.413 and 28/150 is about 0.187. The rounded decimals may not total exactly 1, but 150/150 = 1 exactly.
If the empirical results differ from the theoretical model, does that mean the model is wrong?
No, a gap alone does not prove the model is wrong. Relative frequency is only an estimate, and sampling variability causes some difference even when the model is exactly correct. With 200 trials, a gap like 0.06 versus 0.1 is worth investigating. A gap that persists as more data is collected is much stronger evidence against the model's assumption.

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