Probability Distribution

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).

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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 $x$.

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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.

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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($\lambda=20$).

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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).

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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.

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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.