Sampling Distributions and Estimator Properties Notes for GATE DA: Concepts, Formulas, Worked Examples & Practice

Sampling Distributions and Estimator Properties notes for GATE DA: 23 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

Chapter Roadmap: Sampling Distributions and Estimator Properties

Chapter Journey: Sampling Distributions and Estimator Properties

Sampling Distributions and Chi-Square Results (Current Topic)

Distribution of

\overset{ˉ}{X}

, Chi-Square definition, Sample Variance Identity.

Estimator Bias and Variance Properties

Bias, MSE, Unbiasedness, Consistency, and Efficiency.

Exam Weightage Hint: Both topics are foundational for Hypothesis Testing and Confidence Intervals. Expect questions that test your ability to identify the correct distribution of a given statistic and to prove unbiasedness using these distributions.

Hero Concept: What is a Sampling Distribution?

What is a Sampling Distribution?

A statistic is any function of the sample data that does not depend on unknown parameters. Examples include the sample mean $\overset{ˉ}{X}$ , sample variance $S^{2}$ , and sample proportion $\overset{p}{^}$ .

Because the sample data is random, the statistic itself is a random variable. The probability distribution of this random variable is called its sampling distribution.

Why It Matters

To make inferences about a population, we must know how our statistic behaves across all possible samples. The sampling distribution provides the exact mathematical model for this behavior.

Concept: Distribution of the Sample Mean

Distribution of the Sample Mean

Let $X_{1}, X_{2}, \dots, X_{n}$ be a random sample from a normal distribution $N (μ, σ^{2})$ . The sample mean $\overset{ˉ}{X} = \frac{1}{n} \sum_{i = 1}^{n} X_{i}$ is a linear combination of independent normal random variables. Therefore, $\overset{ˉ}{X}$ is also normally distributed.

Mean

E [\overset{ˉ}{X}] = μ

Variance

V a r (\overset{ˉ}{X}) = \frac{σ ^{2}}{n}

Standardization

Z = \frac{X ˉ - μ}{σ / n} \sim N (0, 1)

Formula: The Chi-Square Distribution

The Chi-Square Distribution

Let $Z_{1}, Z_{2}, \dots, Z_{k}$ be independent random variables, each following a standard normal distribution $N (0, 1)$ . The random variable $Q$ defined as the sum of their squares:

Q = i = 1 \sum k Z_{i}^{2} \sim χ_{k}^{2}

1 Support:

Q > 0

(Strictly non-negative)

2 Mean:

E [Q] = k

3 Variance:

V a r (Q) = 2 k

4 Additivity:

χ_{a}^{2} + χ_{b}^{2} = χ_{a + b}^{2}

(if independent)

More notes in this unit

Sample Mean, Variance and Descriptive Estimation Notes

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Sampling Distributions and Estimator Properties Notes for GATE DA: Concepts, Formulas, Worked Examples & Practice

Sampling Distributions and Estimator Properties notes for GATE DA: 23 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice

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