Lesson 25 of 1524
Discrete random variables
A binomial count has mean np. The trials are independent, and the success probability stays put.
Practice this chapterA discrete random variable has a list of separate values. Its mean is the sum of each value times its probability.
Binomial: a fixed number of independent trials, two outcomes, and the same success probability p each time. The mean is np and the variance is npq, where q = 1 − p. Geometric waits for the first success. Hypergeometric draws without replacement. Poisson counts events over an interval.
Binomial mean
μ = np
Number of trials times the probability of success on each one.
Worked example
A binomial experiment has n = 20 and p = 0.3. What is the mean?
- 1The binomial mean is μ = np. It is the number of trials times the probability of success on one trial.
- 2n = 20 and p = 0.3, so μ = 20 × 0.3.
- 320 × 0.3 = 6.
- 4The 6 is an average over many repetitions of the whole experiment. One run of 20 trials can have more successes than 6 or fewer.
Result: 6
Why. Each trial contributes 0.3 of a success on average. Twenty independent trials with that same probability add to 20 × 0.3 = 6 expected successes. The formula uses p, not the failure probability 0.7.
If the draws are without replacement and the population is small, the trials are not independent, so the model is hypergeometric rather than binomial.
Practice margin
This chapter
A fresh set from this chapter only. Choose 10 or 20. Multiple choice and fill-in, with no repeat inside the set.