1. Bernoulli Distribution
- Also known as a binary distribution
- Used to model the outcome of a single trial or experiment with two possible outcomes (e.g., heads/tails, pass/fail)
Probability mass function: P(X=k) = p^k \ (1-p)^(1-k), where k=0,1 and p is the probability of success
Example: Flipping a coin once. The outcome is either heads (success) or tails (failure). In this case, p=0.5.
2. Binomial Distribution
- Used to model the number of successes in a fixed number of independent trials, each with two possible outcomes
Probability mass function: P(X=k) = C(n,k) \ p^k \* (1-p)^(n-k), where k=0,1,...,n and n is the number of trials
Example: Drawing 5 cards from a standard deck without replacement. The outcome can be the number of hearts drawn (success). In this case, n=5 and p=13/52.
3. Poisson Distribution
- Used to model the number of events occurring in a fixed interval of time or space
Probability mass function: P(X=k) = e^(-λ) \ (λ^k)/k!, where k=0,1,... and λ is the average rate of events
Example: Number of phone calls received by a call center per hour. The outcome can be the number of calls received in an hour.
4. Normal Distribution
- Also known as the Gaussian distribution
- Used to model continuous data that cluster around the mean with a symmetrical bell-shaped curve
Probability density function: f(x) = (1/√(2πσ^2)) \ e^(-(x-μ)^2/(2σ^2)), where μ is the mean and σ is the standard deviation
Example: Heights of adults in a population. The outcome can be any value within the normal range.
5. Exponential Distribution
- Used to model the time between events or failures
Probability density function: f(x) = λ \ e^(-λx), where x>0 and λ is the rate parameter
Example: Time between failures of a machine. The outcome can be any value greater than 0.
These distributions are widely used in statistics to model real-world phenomena, but they have different properties and applications.