2.5.2 Binomial Flashcards
What is a binomial distribution?
A discrete probability distribution that counts the number of successes in a fixed number of independent Bernoulli trials.
What conditions must be met for a binomial distribution?
- A fixed number of trials (n).
- Each trial has only two outcomes (success or failure).
- The probability of success (p) is constant across trials.
- The trials are independent of each other.
How do we denote that a random variable X follows a binomial distribution?
X ∼ Binomial(n, p), meaning X counts the number of successes in n trials with success probability p.
What is the probability mass function (PMF) of a binomial random variable?
The probability of getting exactly k successes in n trials is:
P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}
where \binom{n}{k} is the number of ways to arrange k successes in n trials.
What is the sum of all probabilities in a binomial distribution?
The total probability must always add up to 1, meaning:
\sum_{k=0}^{n} P(X = k) = 1
What is the mean (expected value) of a binomial distribution?
The expected number of successes in n trials is:
E[X] = n p
What is the variance of a binomial distribution?
The formula for variance is:
Var(X) = n p (1 - p)
What happens to the binomial distribution when n = 1?
It becomes a Bernoulli distribution, which is just a single success/failure trial.
What happens if you sum multiple binomial random variables?
The sum of binomial random variables with the same success probability is also binomial, where the new n is the total number of trials.
When can we assume trials are independent in a binomial setting?
When sampling with replacement.
When sampling without replacement from a very large population.
When is the binomial distribution not valid?
When sampling without replacement from a small population, because the probability of success changes with each trial.
