1. What Is a Distribution?

  • A probability distribution describes how the values of a random variable are spread or allocated across possible outcomes.
  • It tells us:
    • which values are possible,
    • how likely each value is.

Mathematically:

  • For a discrete random variable $X$: $P(X = x) = p(x)$
  • For a continuous random variable: $P(a \leq X \leq b) = \int_a^b f(x)\,dx$ where $f(x)$ is the probability density function (PDF).

2. Key Functions of a Distribution

  • PMF (Probability Mass Function) → for discrete variables (e.g., dice).
  • PDF (Probability Density Function) → for continuous variables (e.g., height).
  • CDF (Cumulative Distribution Function): $F(x) = P(X \leq x)$ gives probability that a variable is less than or equal to $x$.

3. Types of Distributions

Discrete Distributions

  • Bernoulli: Success/failure (coin flip).
  • Binomial: # of successes in $n$ trials.
  • Poisson: Count of events in a fixed time/space interval.
  • Geometric: # of trials until first success.

Continuous Distributions

  • Uniform: All values in interval equally likely.
  • Normal (Gaussian): Bell-shaped, common in nature.
  • Exponential: Time until an event occurs.
  • Gamma / Weibull: General lifetime distributions.
  • t-distribution: Heavy-tailed, used in inference.

4. Examples

  • Tossing a fair coin:
    • Distribution = Bernoulli(0.5).
  • Rolling a die:
    • Distribution = Discrete uniform on {1,2,3,4,5,6}.
  • Adult height:
    • Distribution ≈ Normal($\mu=170, \sigma=10$).
  • Number of daily customer arrivals:
    • Distribution ≈ Poisson($\lambda=20$).

5. Why They Matter

  • Distributions let us:
    • Model uncertainty in data.
    • Make predictions about future outcomes.
    • Do statistical inference (hypothesis testing, confidence intervals).
    • Perform simulations (Monte Carlo).

6. In Forecasting

  • Deterministic forecast → single number.
  • Probabilistic forecast → distribution of future values (e.g., “Next week’s demand follows Normal(500, σ=30)”).
  • Quantiles, prediction intervals, and risk measures all come from distributions.

Summary:
A probability distribution describes how likely different values of a random variable are. Distributions can be discrete (Bernoulli, Binomial, Poisson) or continuous (Normal, Exponential, Uniform). They’re essential for quantifying uncertainty, making forecasts, and performing statistical inference.