Pearson eText for Probability and Statistical Inference -- Instant Access (Pearson+)

10th Edition

ISBN: 9780137538461

Author: Robert Hogg, Elliot Tanis

Publisher: PEARSON+

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Estimation

A new framework of data analysis where the new statistical methods are used for interpreting the results and analyzing the data is known as estimation in statistics.

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Chapter 5.9, Problem 3E

Let S 2 be the sample variance of a random sample of size n from N ( μ , σ 2 ) .

Show that the limit, as n → ∞ , of the mgf of S 2 is e σ 2 t , − ∞ < t < ∞ .

Thus, in the limit, the distribution of 2 is degenerate with probability 1 at σ 2 .

Features Features Normal distribution is characterized by two parameters, mean (µ) and standard deviation (σ). When graphed, the mean represents the center of the bell curve and the graph is perfectly symmetric about the center. The mean, median, and mode are all equal for a normal distribution. The standard deviation measures the data's spread from the center. The higher the standard deviation, the more the data is spread out and the flatter the bell curve looks. Variance is another commonly used measure of the spread of the distribution and is equal to the square of the standard deviation.

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Chapter 5.9, Problem 2E

Chapter 5.9, Problem 4E

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Pearson eText for Probability and Statistical Inference -- Instant Access (Pearson+)

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Let S 2 be the sample variance of a random sample of size n from N ( μ , σ 2 ) . Show that the limit, as n → ∞ , of the mgf of S 2 is e σ 2 t , − ∞ &lt; t &lt; ∞ . Thus, in the limit, the distribution of 2 is degenerate with probability 1 at σ 2 .

Solution Summary: The author shows the sample variance S2 from the normal distribution and the random sample size is n.

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Chapter 5 Solutions

Let S 2 be the sample variance of a random sample of size n from N ( μ , σ 2 ) . Show that the limit, as n → ∞ , of the mgf of S 2 is e σ 2 t , − ∞ < t < ∞ . Thus, in the limit, the distribution of 2 is degenerate with probability 1 at σ 2 .