Introduction to Statistical Inference and Data Analysis by Michael W. Trosset

Author:Michael W. Trosset [Trosset, Michael W.]
Language: eng
Format: epub
Published: 2011-02-19T05:00:00+00:00

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CHAPTER 5. QUANTIFYING POPULATION ATTRIBUTES

(b) For every random variable X, the interquartile range of X is greater than the standard deviation of X.

(c) For every random variable X, the expected value of X lies between the first and third quartile of X.

(d) If the standard deviation of a random variable equals zero, then so does its interquartile range.

(e) If the median of a random variable equals its expected value, then the random variable is symmetric.

4. For each of the following random variables, discuss whether the median or the mean would be a more useful measure of centrality: (a) The annual income of U.S. households.

(b) The lifetime of 75-watt light bulbs.

5. The S-Plus function qbinom returns quantiles of the binomial distribution. For example, quartiles of X ∼ Binomial(n = 3; p = 0.5) can be computed as follows:

> alpha <- c(.25,.5,.75)

> qbinom(alpha,size=3,prob=.5)

[1] 1 1 2

Notice that X is a symmetric random variable with center of symmetry θ = 1.5, but pbinom computes q2(X) = 1. This reveals that S-Plus may produce unexpected results when it computes the quantiles of discrete random variables. By experimenting with various choices of n and p, try to discover a rule according to which pbinom computes quartiles of the binomial distribution.

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