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Chapter 6. Continuous Random Variables and Probability Distributions

Unit 6.1. Properties of integrals

Please review the information about distribution functions from Distribution function. The definition of the distribution function given there is general and applies equally to discrete and continuous random variables. It has been mentioned that in case of a discrete variable $latex X$ the argument $latex x$ of its distribution function is usually restricted to the values of $latex X$. Now we are going to look into the specifics of the continuous case.

Integral as area. Let the function $latex f$ be defined on the segment $latex [a,b]$. By an integral

$latex \int\limits_a^b {f(x)dx} \qquad (\textrm{Eq. 6.1})$

we mean the area delimited by the graph of $latex f$, the x-axis and two vertical lines: one at $latex a$ (called a lower limit of integration) and another at $latex b$ (called an upper limit of integration).

Where $latex f$ is negative, the area is counted with a negative sign. Thus, for the function in Figure 6.1 $latex \int\limits_a^b {f(x)dx} = -{A_1} + {A_2}$ where $latex {A_1}$ is the area below the x-axis and above the graph and $latex {A_2}$ is the area above the x-axis and below the graph.

Figure 6.1. Integral as area.

Property 1. An integral does not depend on the variable of integration. To see what the arguments of the integral are, try to see in Figure 6.1 what affects its value. If we change any of $latex a,b,f$, the integral will change. The argument $latex x$ does not affect the value of the integral, even though it is present in the notation of the integral Eq. (6.1). For example, the same integral can be written as $latex \int_a^b {f(t)dt}$ and it will not depend on $latex t$. The variable of integration $latex t$ is included here to facilitate some manipulations with integrals.

Property 2. Integral as a function of a segment. Instead of changing separately $latex a$ or $latex b$ we can think of the whole segment $latex [a,b]$ as changing, and then we can ask what properties the integral has as a function of a segment (when $latex f$ is fixed). Its fundamental and geometrically obvious property is additivity with respect to the domain of integration: if $latex a < c < b$, then

$latex \int\limits_a^b {f(x)dx} = \int\limits_a^c {f(x)dx} + \int\limits_c^b {f(x)dx}$

(see Figure 6.1).

Property 3. Integral of a zero length segment. Since the area of a segment is equal to zero, for any $latex a$ we have

$latex \int\limits_a^a {f(x)dx} = 0. \qquad (\textrm{Eq. 6.2})$

Property 4. Integral as a function of the integrand. We can also ask how the integral changes when the segment $latex [a,b]$ is fixed and the function $latex f$ (the integrand) changes. Let $latex f,g$ be two functions and let $latex c,d$ be two numbers. The function $latex cf(x) + dg(x)$ is called a linear combination of $latex f,g$ with coefficients $latex c,d$. The linearity property states that

$latex \int\limits_a^b {[cf(x) + dg(x)]dx} = c\int\limits_a^b {f(x)dx} + d\int\limits_a^b {g(x)dx}.$

This property is quite similar to linearity of means and its applications are along the same lines but the proof requires a different level of math. This is one of those cases when you can use the cow-and-goat principle: here is a cow and there is a goat; they are quite similar and can be used, to a certain extent, for the same purposes. When it's time to tell you how the cow is different from the goat, I will.

Property 5. Integral preserves order. The order property of integrals is immediate from their interpretation as area: if the graph of the function $latex f$ is above that of the function $latex g$, $latex f(x) \ge g(x)$ for all $latex x \in [a,b]$, then $latex \int\limits_a^b {f(x)dx} \ge \int\limits_a^b {g(x)dx}.$

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