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Chapter 6. Continuous Random Variables and Probability Distributions

Unit 6.3. Mean value of a continuous random variable

This whole section is about the definition of $latex EX$ for a continuous $latex X$. In the preamble of the definition you will see the argument of type used in the times of Newton and Leibnitz, the fathers of calculus. It is still used today in inductive arguments by modern mathematicians.

Approximation of integrals by integral sums. The idea of the next construction is illustrated in Figure 6.2. The segment $latex [a,b]$ is divided into many small segments, denoted

$latex [{t_0},{t_1}],\,\,....,\,\,[{t_{m - 1}},{t_m}] \qquad (\textrm{Eq. 6.5})$

where the leftmost point $latex {t_0}$ coincides with $latex a$ and the rightmost point $latex {t_m}$ is $latex b$. By additivity of integrals, the integral over the whole segment is a sum of integrals over subsegments:

$latex \int\limits_a^b {f(x)dx} = \int\limits_{{t_0}}^{{t_1}} {f(x)dx} + ... + \int\limits_{{t_{m - 1}}}^{{t_m}} {f(x)dx}. \qquad (\textrm{Eq. 6.6})$

Figure 6.2. Approximation of an integral.

When $latex m$ is large and the subsegments are small, for each of them the area $latex \int\limits_{{t_{i - 1}}}^{{t_i}} {f(x)dx}$ of a curvilinear figure can be approximated by the area $latex f({t_i})({t_i} - {t_{i - 1}})$ of a rectangle. Therefore Eq. (6.6) is approximately

$latex \int\limits_a^b {f(x)dx} \approx f({t_1})({t_1} - {t_0}) + ... + f({t_m})({t_m} - {t_{m - 1}}). \qquad (\textrm{Eq. 6.7})$

In particular, if the segment $latex [a,b]$ itself is small, then

$latex \int\limits_a^b {f(x)dx} \approx f(b)(b - a). \qquad (\textrm{Eq. 6.8})$

Preamble. Let $latex X$ be a continuous random variable that takes values in $latex [a,b]$. Let us divide $latex [a,b]$ into equal parts Eq. (6.5). If we have a sample of $latex n$ observations on $latex X$, they can be grouped in batches depending on which subsegment they fall into. Let $latex {n_1}$ be the number of observations falling into the first subsegment $latex [{t_0},{t_1}]$, ..., $latex {n_m}$ the number of observations falling into the last one $latex [{t_{m - 1}},{t_m}]$. All observations falling into $latex [{t_{i - 1}},{t_i}]$ are approximately equal to the right end $latex {t_i}$ of this subsegment. Then $latex {n_1} + ... + {n_m} = n$ and by the mean of grouped data formula (see the fourth line in Eq. (5.2) in General properties of means)

$latex EX \approx \frac{{{n_1}{t_1} + ... + {n_m}{t_m}}}{n}. \qquad (\textrm{Eq. 6.9})$

In the frequentist interpretation, $latex \frac{{{n_i}}}{n}$ approximates the probability that $latex X$ takes values in $latex [{t_{i - 1}},{t_i}]$. By the interval formula Eq. (5.16) (see Distribution function) and approximation Eq. (6.8)

$latex \frac{{{n_i}}}{n} \approx P({t_{i - 1}} < X \le {t_i}) = \int\limits_{{t_{i - 1}}}^{{t_i}} {{p_X}(t)dt}$

$latex \approx {p_X}({t_{i - 1}})({t_i} - {t_{i - 1}}) \qquad (\textrm{Eq. 6.10})$

when the subsegments are short enough. Equations (6.9) and (6.10) imply that

$latex EX \approx {t_1}{p_X}({t_1})({t_1} - {t_0}) + ... + {t_m}{p_X}({t_m})({t_m} - {t_{m - 1}}). \qquad (\textrm{Eq. 6.11})$

From equations (6.7) and (6.11) we see that

$latex EX \approx \int\limits_a^b {t{p_X}(t)dt}$

where the approximation improves as the partitioning into subsegments becomes finer. This motivates the following definition:

Mean of a continuous random variable. The mean of a continuous random variable which takes values in $latex [a,b]$ is defined by

$latex EX = \int\limits_a^b {t{p_X}(t)dt}.$

Extending the density by zero outside $latex [a,b]$ we can with equal success define the mean by

$latex EX = \int\limits_{ - \infty }^{ + \infty } {t{p_X}(t)dt}. \qquad (\textrm{Eq. 6.12})$

The last equation is adopted as a definition for a variable that can take any real values. Compare Eq. (6.12) with the definition of the mean value in the discrete case Eq. (5.1) (see General properties of means): when you pass from the discrete case to the continuous one, you have to replace summation signs with integrals, values $latex {x_i}$ with $latex t$ (because $latex X$ now takes real values) and the probabilities $latex {p_i}$ with the density $latex {p_X}(t)$. The differential $latex dt$ in Eq. (6.12) really means nothing.

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