If events occur at rate λ per unit time, the number of events in a fixed interval follows which distribution?

• A. Uniform(0, λ)
• B. Poisson(λ)
• C. Normal(λ, 1)
• D. Binomial(λ, 0.5)

Answer: (B)

Count of events in a fixed time interval for events that occur independently at a constant average rate per unit time follows a Poisson process. In such a interval, the number of events is Poisson with parameter equal to the rate times the interval length. If the interval is one unit long, that parameter is λ, so the count is Poisson(λ). This makes sense because the events are discrete and occur with a stable average frequency, and the counts add across nonoverlapping subintervals. The probability of observing k events is e^{-λ t} (λ t)^k / k!, with mean and variance both equal to λ t. The other options don’t fit because uniform is continuous, normal is continuous, and binomial requires a fixed number of trials rather than a random count of events in time.