Grades Pre-Calculus & Calculus Advanced StatisticsProbability Distributions

Probability Distributions

📊 Advanced Statistics

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Probability Distributions

Model random phenomena using binomial, normal, and Poisson distributions.

A probability distribution describes how probability is distributed across the values of a random variable. Discrete distributions (like binomial and Poisson) deal with countable outcomes; continuous distributions (like the normal) assign probabilities to intervals.

\text{Binomial: } P(X=k) = \binom{n}{k}p^k(1-p)^{n-k}, \quad \mu = np,\; \sigma^2 = np(1-p)

\text{Poisson: } P(X=k) = \frac{\lambda^k e^{-\lambda}}{k!}, \quad \mu = \sigma^2 = \lambda

Binomial Distribution

Fixed n trialsEach trial: success (p) or failure (1−p)Trials are independentCount the number of successes X

Poisson Distribution

Events occur at average rate λ per intervalEvents are independentApproximates binomial when n large, p smallCount rare events: calls/hour, typos/page

Use Binomial when n and p are given. Use Poisson when only the average rate λ is given.

\text{Normal: } f(x) = \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}

🎲Binomial Calculation

10 fair coin flips. P(exactly 3 heads) = C(10,3)·(0.5)³·(0.5)⁷ = 120·(1/1024) ≈ 0.117.

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Remember This!

Normal approximation to the binomial is valid when np ≥ 10 and n(1−p) ≥ 10. Apply the continuity correction: P(X ≤ k) ≈ P(Z ≤ (k + 0.5 − np)/√(np(1−p))).

✏️ Try It!

A fair coin is flipped 8 times. What is P(exactly 2 heads)?

Practice with CalcVerse

🎲Dice & Coin Simulator 🔔Normal Distribution & Z-Score

Take Quiz 📝 — 25 Questions