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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 provides the average value of outcomes, while the standard deviation measures the extent of variability or dispersion from this mean. They are computed using the respective formulae: \(E(X) = \sum x_i P(x_i)\) for mean and \(蟽 = 鈭歏ar(X)\) for standard deviation where \(Var(X) = \sum (x_i - 渭)^2 * P(x_i)\).

Step by step solution

01

Mean of a Discrete Random Variable

A mean of a discrete random variable, often known as the expected value, is a measure of the 'central tendency' of the random variable. It provides the average value of outcomes if the experiment is repeated many times. It is computed as the sum of all possible values each multiplied by the probability of its occurrence. Mathematically, it is represented as \(E(X) = \sum x_i P(x_i)\) where \(x_i\) is each value the variable can take and \(P(x_i)\) is the respective probability.
02

Standard Deviation of a Discrete Random Variable

The standard deviation is a measure of the range of the values a random variable can take. In essence, it tells us how spread out the data from the mean is. It is the square root of the variance. The variance, represented as \(Var(X)\), is calculated as \(Var(X) = E((X 鈥� 渭)^2)= \sum (x_i - 渭)^2 * P(x_i)\) where 渭 is the mean of the discrete random variable. The standard deviation (蟽) is then obtained by taking the square root of the variance, hence \(蟽 = 鈭歏ar(X)\).
03

Importance of Mean and Standard Deviation

The mean gives a central value for the data set while the standard deviation measures the extent of variability or dispersion from this mean. Hence, both measures are useful for summarizing a set of data. They provide insights into what values are 'normal' or 'expected' and how much deviation from these 'expected' values is common.

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