Binomial Distribution

Slud's inequality converts a specified binomial probability into a normal-tail lower bound.

Programming

Interactive lab

Try it: Binomial Distribution from Bernoulli Trials

A binomial variable counts the successes in n independent Bernoulli(p) trials: Pascal's triangle counts the trial sequences with k successes, , and a seeded simulation of many runs lands on that PMF.

How it works

One run: n independent trials, each a success with probability p; X is the number of successes.
Pascal's rule C(r, k) = C(r − 1, k − 1) + C(r − 1, k) counts the success/failure paths that end with k successes.
Every such path has probability , so .
Mean np and variance np(1 − p), checked against Σ k·P(X = k).
Repeat the n trials many times (seeded) and overlay the observed frequencies on the exact PMF.

Default run (35 steps): X ~ Binomial(n = 10, p = 0.3): X counts the successes in 10 independent Bernoulli(0.3) trials, so X can be 0, 1, …, 10. … 200 runs: simulated mean 2.98 vs np = 3, simulated variance 2.11 vs np(1 − p) = 2.1. Largest gap is at k = 5: 0.16 simulated vs 0.103 exact.

Simplified: n ≤ 20, p in steps of 0.01, up to 1,000 simulated runs from a seeded pseudo-random generator (mulberry32), so a seed always gives the same runs.

Educational simulation

Loading the simulation…

From Trials to a Count

A binomial variable counts successes across independent Bernoulli trials. Each Bernoulli trial represents a success-or-failure experiment, and the binomial count is formed by adding the individual trial variables. The independence condition matters because the trials must be independent of one another for this construction.

How do individual success and failure trials combine to produce a binomial count?

Definition

A binomial variable is the sum of independent Bernoulli trials. The resulting value records how many successes occur across those trials.

Slud's Parameter Condition

Slud's inequality is used to convert a specified binomial probability into a lower bound involving a normal-tail probability. The parameter relationship required in this section is p = (1 − ε) / 2. Before applying the inequality, check that the binomial parameter p has this relationship to ε.

Formula

How does a specified binomial probability become a normal-tail lower bound, and where do the parameters enter?

The inequality has two conceptual stages. First, identify the binomial probability of interest and verify the relationship between p and ε. Second, replace that specified probability with a lower-bound expression involving a normal tail. The source material states this conversion but does not specify a particular event, threshold, or expanded normal-tail formula, so those details must come from the exact version of Slud's inequality being used.

Parameter Substitution

Worked example

Checking the Slud Relationship

Suppose ε = 0.2. Determine the value of p required by the parameter relationship used in Slud's inequality.

1
Write the relationship
Use p = (1 − ε) / 2.
2
Substitute ε
Replace ε with 0.2, giving p = (1 − 0.2) / 2.
3
Simplify
The required value is p = 0.4.
4
Interpret the result
A binomial probability associated with this p value satisfies the stated parameter relationship. The exact event and normal-tail expression still depend on the specific statement of Slud's inequality being applied.

Result

For ε = 0.2, the required binomial parameter is p = 0.4.

1
Identify the binomial probability that the problem asks about.
2
Check the required relationship p = (1 − ε) / 2.
3
Insert the specified event and parameters into the stated version of Slud's inequality.
4
Read the resulting normal-tail expression as a lower bound for the original binomial probability.

A Different Construction: χ2

A χ2 variable is constructed differently from a binomial variable. Instead of adding Bernoulli success-or-failure trials, form the squares of independent standard normal variables and add those squared values together.

How do squared independent standard-normal variables combine to form a χ2 variable with k degrees of freedom?

Definition

A χ2 variable with k degrees of freedom is the sum of k squared independent standard normal variables.

Formula

Key point

The number of squared terms determines the degrees of freedom. With k squared independent standard-normal terms, the χ2 variable has k degrees of freedom and mean k.

Common Interpretation Errors

Common mistake
Treating a binomial variable as a single trial.
What it looks like
Calling one success-or-failure variable the entire binomial count.
Why it fails
A binomial variable counts successes across a collection of independent Bernoulli trials.
Do this instead
Describe the binomial variable as the sum of the individual independent trials.
Common mistake
Forgetting the condition p = (1 − ε) / 2 when discussing Slud's inequality.
What it looks like
Applying the comparison without checking how p and ε are related.
Why it fails
The stated parameter relationship is a required condition for the inequality in this section.
Do this instead
Verify the relationship before converting the specified binomial probability into a normal-tail lower bound.
Common mistake
Confusing the binomial construction with the χ2 construction.
What it looks like
Describing a χ2 variable as a count of successes.
Why it fails
A χ2 variable is formed by summing squared independent standard normal variables.
Do this instead
Track both the source variables and the operation: standard normals are squared, then the squares are added.
Common mistake
Using the number of normal variables as an unrelated label.
What it looks like
Saying that k degrees of freedom does not affect the mean.
Why it fails
The number of squared terms is k, and the mean of the resulting χ2 variable is k.
Do this instead
Connect the number of squared terms, the degrees of freedom, and the mean.

Practice Check

MEDIUM

A binomial variable uses a parameter p and Slud's inequality requires p = (1 − ε) / 2. If p = 0.3, determine ε. Then describe how a specified probability for this binomial variable is transformed by Slud's inequality.

Hints

Start with 0.3 = (1 − ε) / 2.
After finding ε, state the conversion in words: the binomial probability becomes a lower bound involving a normal tail.
Do not invent a threshold or a more detailed tail formula unless it is supplied by the specific version of the inequality.

Your attempt (not saved)

EASY

Suppose a χ2 variable is formed from k squared independent standard normal variables. What are its degrees of freedom and its mean?