LGT5101 Chap.5 Discrete Random Variables and Binomial Models
Discrete Random Variables and Binomial Models
Put probability on countable values
A discrete random variable assigns numerical values to experimental outcomes. Its probability mass function lists P(X=x) for every possible value. Each mass must lie between zero and one and the masses must sum to one. A bar at a value represents exact probability, unlike a histogram interval.
Define the random variable in words so a numerical result remains attached to its experiment and unit.
Separate exact from cumulative questions
The cumulative distribution F(k) adds mass through k and answers P(X less than or equal to k). For a discrete variable, strict and inclusive thresholds differ because a point can carry positive probability. Thus P(X greater than k)=1-F(k), while P(X at least k)=1-F(k-1).
A valid cdf never decreases and finishes at one; each jump equals the pmf at that value.
Build expectation and variance transparently
Expectation sums xP(X=x) and represents a probability-weighted long-run centre, not necessarily an attainable outcome. Variance sums probability-weighted squared distance from that centre, or equivalently uses E(X squared)-[E(X)] squared. Standard deviation restores the units of X.
A table containing x, probability, weighted value and weighted square makes the arithmetic auditable and exposes missing probability mass before it contaminates later summaries.
Earn the binomial shortcut
A binomial model requires a fixed number of trials, two outcomes after any grouping, a constant success probability and independent trials.
Then an exact probability combines a choice count with success and failure powers, while mean and variance reduce to np and np(1-p). Sampling without replacement from a small population can violate independence. For capacity decisions, translate at most, more than and at least into the correct cdf boundary, and verify both the first passing integer and the preceding failing integer when a minimum is requested.
What this chapter covers
- 01
Discrete random variables
- 02
Probability mass functions
- 03
Cumulative distributions
- 04
Expectation
- 05
Variance and standard deviation
- 06
Binomial conditions
- 07
Exact binomial mass
- 08
Cumulative binomial risk
Worked example · free
Damage risk in a parcel audit
- 3Verify and parameterise the binomial model.
- 4Calculate the exact probability.
- 3Calculate and interpret mean and spread.
Key terms
- Random Variable
- A numerical value assigned to the outcome of an experiment.
- Probability Mass
- The probability that a discrete random variable equals one possible value.
- Cumulative Distribution
- The probability that a random variable is at or below a threshold.
- Expected Value
- The probability-weighted long-run average of a random variable.
- Binomial Distribution
- The distribution of a success count from fixed independent trials with constant probability.
Discrete Random Variables and Binomial Models FAQ
How do I check whether a pmf is valid?
Verify that every listed probability lies between zero and one and that the probabilities across all possible values sum to one. Both conditions are necessary. A table whose entries are individually plausible but total 0.90 is missing probability mass and cannot support an expectation.
What does a non-integer expected count mean?
Expectation is the average count over many repetitions of the experiment, not a prediction that one run produces a fraction of an item. An expectation of 0.48 damaged parcels means the total damage count would average 0.48 across many six-parcel audits under the model.
How do I prove a production count is minimal?
Compute the fulfilment probability for candidate integer counts until the target is first reached. Report the first passing count and also check the previous count fails. One feasible value proves adequacy, while the neighbouring failure proves that no smaller count meets the stated target.
Exam move
For each binomial problem, write four conditions before any function: fixed trials, two outcomes, constant p and independence. Practise translating exactly, at most, more than and at least into pmf, cdf or complement forms. When a question asks for a minimum capacity, build a short table of candidate n values and verify the first crossing with its immediate predecessor.
Build one complete pmf table and add columns for xP(X=x) and x squared times P(X=x). Check total mass, calculate expectation and variance by both equivalent formulas, and confirm the answers match the binomial shortcuts when the table is binomial. Convert the pmf to a cdf and recover each mass from successive jumps. In Excel, pair every BINOM.DIST expression with the event it represents and explain the TRUE or FALSE flag.
Finally, change one condition—sampling without replacement from a small batch, varying p or stopping early—and explain precisely why the original binomial model no longer describes the experiment.
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