Chapter 5 – Discrete Random Variables (Review Flashcards)

Thirty foundational Q&A flashcards covering definitions, formulas, conditions, and key examples for discrete random variables, their probability distributions (binomial, hypergeometric, Poisson), and related concepts such as expectation, variance, and distribution relationships.

What is a random variable (informal definition)?

A quantitative variable whose observed value is partly determined by chance.

What distinguishes a discrete random variable from a continuous one?

Discrete RVs have a countable set of possible values; continuous RVs can take any value in an interval (uncountably infinite).

Give one example of a discrete random variable.

The number of heads in 10 coin tosses (any correct discrete example earns full credit).

Give one example of a continuous random variable.

The time until a light bulb fails (any correct continuous example earns full credit).

State the two conditions that every discrete probability distribution must satisfy.

1) all probabilities are between 0 and 1, 2) the sum of all probabilities=1

What do we call the function p(x) that yields the probabilities for a discrete random variable?

The probability mass function (pmf).

List the three conditions that define n independent Bernoulli trials (binomial setting).

1) Fixed number n of independent trials; 2) Each trial results in success or failure; 3) P(success) = p is the same on every trial.

When deciding “Binomial or Not,” what key feature usually fails for sampling without replacement from a small population?

Independence of trials (probability of success changes after each draw).

Describe the scenario that produces a hypergeometric distribution.

Drawing a sample of size n without replacement from a population containing a successes and N – a failures.

Rule of thumb: When can a binomial approximate a hypergeometric well?

When the sample size n is no more than about 5 % of the population size N (so removal has little effect on probabilities).

In words, when is the Poisson distribution an appropriate model?

When events occur randomly and independently over time/space and at a constant average rate.

State the two key assumptions that justify a Poisson model for counts in time.

1) Events occur independently; 2) The probability of an event in a given interval is proportional to the length of the interval (constant rate).

Explain the relationship between the binomial and Poisson distributions.

As n → ∞ and p → 0 with λ = np held constant, Binomial(n, p) converges to Poisson(λ); hence Poisson(λ) approximates Binomial(n, p) when n is large and p is small.