Probability Distributions — Normal, Binomial, and Poisson Distributions

Types of Probability Distributions

Discrete Probability Distribution: X takes countable values (e.g., number of heads in coin flips)
Continuous Probability Distribution: X takes continuous values (e.g., height, weight)

Binomial Distribution

Conditions:

n independent trials
Each trial results in exactly two outcomes: success (p) or failure (1−p)
Probability of success p is the same for every trial

\begin{aligned} &X ~ B(n, p) \\ P(X &= k) = C(n,k) \times pᵏ \times (1−p)ⁿ⁻ᵏ \\ \text{Expected value} E(X) &= \text{np} \\ \text{Variance Var}(X) &= \text{np}(1−p) \end{aligned}

Example: Rolling a die 10 times — number of times a “1” appears
→ B(10, 1/6), E(X) = 10/6 ≈ 1.67

When n is large and p is small: Approximate with the Poisson distribution (λ = np)

Poisson Distribution

Distribution of the number of events occurring in a fixed unit of time (or space).

Conditions:

Events occur independently and randomly
Average rate λ is constant

\begin{aligned} &X ~ \text{Poisson}(\lambda) \\ P(X &= k) = (e⁻\lambda \times \lambdaᵏ) / k! \\ E(X) &= \text{Var}(X) = \lambda \end{aligned}

Applications: Number of customers arriving per hour, number of defects per 1,000 items (e.g., manufacturing quality control, rare disease incidence)

Normal Distribution

The most important continuous probability distribution in statistics.

\begin{aligned} &X ~ N(\mu, \sigma^2) \\ f(x) &= (1/\sqrt(2\pi\sigma^2)) \times \text{exp}(−(x−\mu)\frac{^2}{2}\sigma^2) \end{aligned}

Properties:

Symmetric about the mean μ
Mean = Median = Mode
Wider spread as σ (standard deviation) increases
Total area under the curve = 1

Empirical Rule (68–95–99.7 Rule)

μ ± 1σ range: approximately 68.27%
μ ± 2σ range: approximately 95.45%
μ ± 3σ range: approximately 99.73%

Standard Normal Distribution and Z-Scores

Standardization:

\begin{aligned} Z &= (X − \mu) / \sigma \\ &Z ~ N(0, 1) \quad \text{(standard normal distribution)} \end{aligned}

Interpretation of Z-score: How many standard deviations above or below the mean the original value falls.

Example: An exam with mean 75 and SD 10
A student scoring 95: Z = (95 − 75) / 10 = 2.0 → 2 standard deviations above mean = approximately top 2.3%

Applications:

Comparing variables with different units (e.g., SAT vs ACT scores)
Outlier detection (|Z| > 3: extreme value)
Probability calculations using the standard normal table (Z-table)

Exponential Distribution

Distribution of the waiting time between events in a Poisson process.

\begin{aligned} &X ~ \text{Exp}(\lambda) \\ E(X) &= 1/\lambda \quad \text{(mean waiting time)} \\ \text{Var}(X) &= 1/\lambda^2 \\ \text{Memoryless Property}: P(X > s+t | X > s) &= P(X > t) \end{aligned}

Applications: Electronic device lifetimes, customer service wait times, time between system failures

Key Concept Cards

68–95–99.7 Rule (Empirical Rule) ★★★★★ : μ±1σ = 68%, μ±2σ = 95%, μ±3σ = 99.7%. The foundational benchmark for quality control and statistical testing. Memory tip: 1σ=68, 2σ=95, 3σ=99.7

Z-Score (Standardization) ★★★★★ : Z = (X−μ)/σ. Converts raw scores to the standard normal distribution. Enables comparison of data with different units. Memory tip: Z = deviation ÷ standard deviation

Binomial Expected Value ★★★★☆ : E(X) = np, Var(X) = np(1−p). n trials, probability of success p. Memory tip: expected value = number of trials × probability of success

Practice Questions

Q. If test scores follow N(70, 100), what is the probability of scoring 80 or higher? (Z = 1.0 → cumulative probability 84.1%)

Z = (80−70)/10 = 1.0. P(X ≥ 80) = 1 − 0.841 = 0.159 ≈ 15.9%.

Q. What is the relationship between the Poisson and binomial distributions?

When n is large (many trials) and p is small (low success rate), the binomial distribution B(n,p) can be approximated by a Poisson distribution with λ = np. Example applications: modeling defects, rare disease events, or call center arrivals.