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Chapter 5: Problem 22

Briefly explain the concept of the mean and standard deviation of a discrete random variable.

Short Answer

Expert verified
The mean, or expected value, of a discrete random variable is like an average outcome calculated by summing products of each outcome and its associated probability while dividing by the total number of outcomes. Standard deviation is a statistical measurement in dispersion, quantifying the amount of variation or dispersion. A lower standard deviation indicates that the values tend to be closer to the mean of the set, while a high standard deviation indicates that the values are spread out over a wider range.

Step by step solution

01

Mean of a Discrete Random Variable

The mean, often referred to as the expected value, of a discrete random variable represents its average outcome. It is calculated by summing over all possible outcomes, the product of each outcome and its associated probability. In mathematical terms, if \(X\) is a discrete random variable with probability mass function \(p(x)\), then the mean or expected value of \(X\), denoted by \(E(X)\), is given by \(E(X) = \sum_{all x} x*p(x)\)
02

Interpretation of the Mean

The mean is a measure of the 'center' of the probability distribution. It provides an average value that the outcomes of the random variable tend to cluster around. However, it doesn't provide information about the variability or 'spread' of the values
03

Standard Deviation of a Discrete Random Variable

The standard deviation is a measure of the amount of variation or dispersion in the set of values of a discrete random variable. It鈥檚 calculated as the square root of the variance, which is the average of the squared differences of each value from the mean. The mathematical formula for the standard deviation \(\sigma\) of a discrete random variable \(X\) is given by, \(\sigma = \sqrt{E[(X- E(X))^2]}\)
04

Interpretation of the Standard Deviation

A lower standard deviation means that the values are more tightly packed around the mean, while a higher standard deviation indicates that the values are spread out over a wider range. Thus, it represents the variability of the probability distribution.

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Most popular questions from this chapter

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