Standard Distributions: Normal, Exponential, Poisson and Chi-Square Notes for GATE DA: Concepts, Formulas, Worked Examples & Practice

Standard Distributions: Normal, Exponential, Poisson and Chi-Square notes for GATE DA: 31 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

Chapter Roadmap: Standard Distributions

Your Journey Through This Chapter

Poisson and Standard Normal Properties
Counting rare events and the universal continuous ruler.

Exponential Distribution
Time until the next event and the memoryless property.

Normal and t-Distribution Comparisons
Density functions, CDFs, and comparing the bell curves.

The Two Pillars: Counting Events vs Measuring Continuously

This chapter focuses on the most important continuous and discrete distributions in probability. We start with two foundational pillars:

Poisson Distribution

Models the number of discrete events occurring in a fixed interval of time or space.

Standard Normal Distribution

Models continuous measurements that cluster symmetrically around a mean.

Understanding these two distributions provides the foundation for almost all other distributions you will encounter.

Poisson Distribution: The Law of Rare Events

The Poisson distribution models the number of discrete events occurring in a fixed interval of time, space, or volume, given a constant average rate.

Probability Mass Function (PMF):

P (X = k) = \frac{e ^{- λ} λ ^{k}}{k !}

where $k = 0, 1, 2, \dots$ and $λ > 0$ is the average rate.

Key Characteristics:

Discrete: The random variable $X$ takes only non-negative integer values.
Single Parameter: Completely defined by $λ$ (the expected number of events).
Independent Events: The occurrence of one event does not affect the probability of another.

Poisson: The Magic of Equal Mean and Variance

The most powerful feature of the Poisson distribution is the relationship between its mean and variance.

E [X] = λ

V a r (X) = λ

Why this matters: If a problem states that the mean and variance of a discrete random variable are equal, it is a massive hint that the variable follows a Poisson distribution. This equality is unique to Poisson among common discrete distributions.

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Standard Distributions: Normal, Exponential, Poisson and Chi-Square Notes for GATE DA: Concepts, Formulas, Worked Examples & Practice

Standard Distributions: Normal, Exponential, Poisson and Chi-Square notes for GATE DA: 31 study cards covering concepts, formulas, shortcuts and exam traps, p

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