Poisson Distribution Calculator
Statistics, Probability
Intro: Computes Poisson probabilities for counts of rare events given an average rate λ over a fixed interval.
Worked example
- A call center receives 4.5 calls per minute on average. What is P(X=3 calls in a minute)?
- The Poisson formula is $P(X=k)=\dfrac{\lambda^k e^{-\lambda}}{k!}$.
- Here $\lambda = 4.5$ (average calls per minute) and $k=3$ (we want exactly 3 calls).
- Substitute into the formula: $P(X=3)=\dfrac{4.5^3 e^{-4.5}}{3!}$.
- Compute the numerator power: $4.5^3 = 4.5\cdot4.5\cdot4.5 = 91.125$.
- Compute the factorial: $3! = 3\cdot2\cdot1 = 6$.
- So $P(X=3)=\dfrac{91.125}{6}e^{-4.5} \approx 15.1875\cdot e^{-4.5}$.
- Evaluate $e^{-4.5} \approx 0.0111$, so $P(X=3) \approx 15.1875\cdot0.0111 \approx 0.169$ (about 16.9%).
- Answer: $\boxed{P(X=3) \approx 0.169}$, so there is about a $16.9\%$ chance of exactly 3 calls in a minute.
FAQs
Can I use non-integer λ?
Yes. λ is an average rate and may be any positive real number, even if counts X are integers.
Why choose MathGPT?
- Get clear, step-by-step solutions that explain the “why,” not just the answer.
- See the rules used at each step (power, product, quotient, chain, and more).
- Optional animated walk-throughs to make tricky ideas click faster.
- Clean LaTeX rendering for notes, homework, and study guides.
How this calculator works
- Type or paste your function (LaTeX like
\sin,\lnworks too). - Press Generate a practice question button to generate the derivative and the full reasoning.
- Review each step to understand which rule was applied and why.
- Practice with similar problems to lock in the method.