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HPAS MainsEconomics (Optional)Science and TechnologyPractice question

Probability Distributions: Binomial, Poisson, and Normal

What is a probability distribution? Explain Binomial, Poisson, and Normal distributions.

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Define what constitutes a probability distribution for both discrete and continuous random variables. Systematically explain Binomial, Poisson, and Normal distributions by stating their probability functions, key parameters, underlying assumptions, and moments. Conclude by outlining the limiting relationships and convergence among these distributions.

Model answer

474 words

Introduction

A probability distribution mathematically specifies the likelihood of various outcomes of a random variable. Depending on whether the variable is discrete or continuous, it is characterized by a Probability Mass Function (PMF) or a Probability Density Function (PDF), fulfilling fundamental axiomatic conditions of non-negativity and total probability summing or integrating to unity.

1. Binomial Distribution (Discrete)

The Binomial distribution models the number of successes in a fixed number of independent and identical Bernoulli trials.

  • Probability Mass Function (PMF): P(X = x) = nCx px qn-x, where x = 0, 1, 2, ..., n and q = 1 - p.
  • Assumptions: The experiment consists of n finite, identical, and mutually independent trials, with each trial resulting in one of two mutually exclusive outcomes ('success' or 'failure') and a constant probability of success (p).
  • Key Properties: Mean = np, Variance = npq. The distribution exhibits positive skewness if p < 0.5, negative skewness if p > 0.5, and is symmetric if p = 0.5, with skewness given by (q - p) / √(npq).

2. Poisson Distribution (Discrete)

The Poisson distribution models the number of occurrences of an event within a fixed interval of time, space, or volume.

  • Probability Mass Function (PMF): P(X = x) = (e-λ λx) / x!, where x = 0, 1, 2, ... ∞, governed by the rate parameter λ > 0.
  • Assumptions and Applications: Events occur independently at a constant average rate, and simultaneous occurrences in an infinitesimally small interval are negligible. It is extensively applied to model rare events (e.g., annual frequency of major landslides in Kinnaur or system arrival queues).
  • Key Properties: Equidispersion property where Mean = Variance = λ. The skewness is 1 / √λ.
  • Limiting Form: It emerges as the limiting form of the Binomial distribution when n → ∞, p → 0, such that np remains finite and equals λ.

3. Normal Distribution (Continuous)

The Normal (or Gaussian) distribution is the pre-eminent continuous probability distribution across theoretical and applied statistics.

  • Probability Density Function (PDF): f(x) = (1 / (σ√(2π))) exp[-0.5 ((x - μ) / σ)2], for -∞ < x < ∞, characterized by mean μ and variance σ2.
  • Key Properties: Symmetrical and bell-shaped curve where Mean = Median = Mode. Skewness = 0, and Kurtosis (β2) = 3 (mesokurtic). The total area under the curve is 1.
  • Limiting Relationships: Both the Binomial and Poisson distributions asymptotically converge to the Normal distribution as sample size n → ∞ or λ → ∞, forming the foundational basis of the Central Limit Theorem.

Conclusion

Probability distributions serve as the foundational bedrock for statistical inference and empirical modeling. While Binomial and Poisson distributions effectively handle counting processes and discrete event frequencies, the Normal distribution provides the analytical bridge for continuous data and large-sample approximations across diverse scientific disciplines.

Key facts to remember

definition
Probability Mass Function (PMF)

A mathematical function that gives the probability that a discrete random variable is exactly equal to some value, satisfying non-negativity and summing to 1 across all possible values.

definition
Probability Density Function (PDF)

A function representing the relative likelihood for a continuous random variable to take on a given value; the integral over any interval yields the probability of the variable falling within that interval.

example
Poisson Distribution in Hazard Analysis

Modeling the annual frequency of rare natural hazards, such as localized landslides in high-altitude Himalayan terrains like Kinnaur, where events occur randomly and independently over a fixed continuous time continuum.

Frequently asked questions

How does the Binomial distribution relate to the Poisson and Normal distributions?

The Binomial distribution converges to the Poisson distribution when trials n are very large and probability p is very small (np = λ). It approximates the Normal distribution when n is large and neither p nor q is close to zero.

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