Probability Distributions

Probability distributions describe the likelihood of different outcomes. Discrete distributions include Binomial (number of successes in n trials), Poisson (number of events in an interval), and Geometric (waiting time). Continuous distributions include Uniform, Exponential, Normal, and Weibull. The probability mass function (PMF) applies to discrete variables, while the probability density function (PDF) applies to continuous variables. The cumulative distribution function (CDF) gives P(X ≤ x).

Governing FormulaBinomial: P(X=k) = C(n,k) × p^k × (1-p)^(n-k), Poisson: P(X=k) = (λ^k × e^(-λ))/k!

Knowledge Check

10 Questions

1.A fair coin is flipped 5 times. What is the probability of getting exactly 3 heads?

2.What is the mean of a Binomial distribution with n = 20 and p = 0.4?

3.A Poisson distribution with λ = 4 is used to model the number of arrivals per hour. What is the probability of exactly 2 arrivals? (e^(-4) ≈ 0.0183)

4.The variance of a Binomial distribution with n = 15 and p = 0.6 is:

5.In a Poisson distribution, the variance is equal to:

6.The exponential distribution is commonly used to model:

7.What is the mean of a Uniform distribution on the interval [a, b]?

8.A continuous random variable has a probability density function (PDF) that must satisfy:

9.The binomial distribution is used for:

10.A Poisson process has a rate of 5 events per hour. What is the probability of no events in 30 minutes? (e^(-2.5) ≈ 0.0821)