 maths > calculus-limits

Expected Value of a Function

what you'll learn...

Expected Value of Functions

»  A function evaluate to 00$\frac{0}{0}$ at x=a$x = a$
→  Considering that functions model real world quantities, in setting x=a$x = a$ there can be very small error.

→  Considering functions in abstraction, the value at x=a$x = a$ can be assumed to be that of x=a±δ$x = a \pm \delta$ with $\delta$ close to $0$

»  If a function is evaluated to $\frac{0}{0}$ at $x = a$
→  expected value can be that of $x = a - \delta$ : left-hand-limit
→  expected value can be that of $x = a + \delta$ : right-hand-limit

»  $f \left(x\right) = \frac{\left(x + 2\right) \left(x - 2\right)}{x - 2}$
→  $x - 2$ can be canceled for all values of $x$ except for $x = 2$ (as $0$ cannot be canceled).
→  function at value $f \left(x\right) {|}_{x = 2} = \frac{0}{0}$ :
→  left hand limit ${\lim}_{x \to 2 -} f \left(x\right) = 2 - \delta + 2 = 4$
→  right hand limit ${\lim}_{x \to 2 +} f \left(x\right) = 2 + \delta + 2 = 4$

indeterminate value

Some examples of functions evaluating to indeterminate values are

$\frac{\sin x}{x}$ when $x = 0$

$\frac{{e}^{x} - 1}{x}$ when $x = 0$

$\frac{f \left(x + \delta\right) - f \left(x\right)}{\delta}$ when $\delta = 0$

what a function means?

It is noted that function provide a mathematical model to applications.

Examples:

Shapes - like circumference of circle

Motion - like speed of a ball thrown

System - like heat emitted in chemical reactions

application scenario of the function

In such applications we come across functions that have indeterminate values at certain values for the argument variable. The motivation in applied mathematics is to solve such functions having indeterminate values. In such systems, setting a parameter to an absolute value is not practically possible. This is examined further in following.

The motivation in abstract mathematics is to find what is the expected value of the function where it has indeterminate value. It is known that indeterminate value $\frac{0}{0}$ possibly takes a value depending on the problem. Can that be determined?

Note: Apart from evaluating a function at an input value, there is another motivation to the theory in discussion. It is covered in the topic on 'continuity of functions'.

Given that function $f \left(x\right)$ evaluates to indeterminate value at $x = a$. The approach to finding the expected value is
•  If this represents an application with some parameter $x = a$, the quantity $x$ cannot be exactly set to value $a$ in practical situations. There will be a small error to value of $x$.

•  circle of radius $3$cm: there will be a tiny error; actual radius is $3.00034$cm.

•  Ball thrown with $2$m/sec velocity : there will be a small error; actual velocity is $1.99948$m/sec

•  If it is abstract, is the function defined at input values very close to $x = a$? Then those values can be used to find expected value of $f \left(a\right)$.

In statistics and probability another form of expected value is studied. It has a different domain and definition. Here, the phrase "expected value" is used to denote the meaning as per plain English. We expect the value to be something because, the nearby values are some value.

close to the position

If a function evaluates to indeterminate value at an input value, then examine the nearby input values close to the input value.

Given that function $f \left(x\right)$ has indeterminate value at $x = a$. To evaluate the possible value of $f \left(x\right) {|}_{x = a}$, we examine values close to $x = a$.

$f \left(x\right) = \frac{{x}^{2} - 1}{x - 1}$ evaluates to indeterminate value at $x = 1$. The values close to $x = 1$ are

at $x = 0.999999$

at $x = 1.000001$

smaller and larger

$f \left(x\right) = \frac{{x}^{2} - 1}{x - 1}$ evaluates to indeterminate value at $x = 1$. But
$f \left(1.1\right) = 2.1$
$f \left(0.9\right) = 1.9$
$f \left(1.001\right) = 2.001$
$f \left(0.99999\right) = 1.99999$
Note that as the value of $x$ is closer to $1$, the $f \left(x\right)$ is closer to $2$.

$f \left(x\right) = \frac{{x}^{2} - 1}{x - 1}$ evaluates to indeterminate value at $x = 1$.

The values closer to $x = 1$ can be either

•  smaller : $.9999 , .9999999$, etc.

•  larger : $1.0001 , 1.0000001$, etc.

This makes two classes of input values for which the function is evaluated.

The two classes are named as given below.

The values smaller than $x = 1$ is referred as 'left-hand-side' and

The values larger than $x = 1$ is referred as 'right-hand-side'

right and left

Given that function $f \left(x\right)$ evaluates to indeterminate value at $x = a$. To evaluate the expected value of $f \left(x\right) {|}_{x = a}$, we examine values close to $x = a$;

•  at right-hand-side $x = a + \delta$

•  at left-hand-side $x = a - \delta$

$f \left(x\right) = \frac{\textcolor{\mathrm{de} e p s k y b l u e}{{x}^{2} - 1}}{\textcolor{c \mathmr{and} a l}{x - 1}}$ evaluates to indeterminate value at $x = 1$. Evaluating right-hand-side at $x = 1 + \delta$

$f \left(1 + \delta\right)$
$\quad \quad = \frac{\textcolor{\mathrm{de} e p s k y b l u e}{{\left(1 + \delta\right)}^{2} - 1}}{\textcolor{c \mathmr{and} a l}{1 + \delta - 1}}$
$\quad \quad = \frac{\textcolor{\mathrm{de} e p s k y b l u e}{1 + 2 \delta + {\delta}^{2} - 1}}{\textcolor{c \mathmr{and} a l}{\delta}}$
$\quad \quad = \frac{\textcolor{\mathrm{de} e p s k y b l u e}{\left(2 + \delta\right) \delta}}{\textcolor{c \mathmr{and} a l}{\delta}}$
$\quad \quad = \textcolor{\mathrm{de} e p s k y b l u e}{2 + \delta}$
$\quad \quad = 2 \quad \quad \quad \quad$ as $\delta \cong 0$

Note that in one of the steps $\delta$ is canceled. That is not same as canceling $0$ as the $\delta \cong 0$. At the same time, in the last step $\delta$ is substituted as $0$, as delta is negligibly small compared to 2.

$f \left(x\right) = \frac{{x}^{2} - 1}{x - 1}$ evaluates to indeterminate value at $x = 1$. Evaluating left-hand-side at $x = 1 - \delta$

$f \left(1 - \delta\right)$
$\quad \quad = \frac{\textcolor{\mathrm{de} e p s k y b l u e}{{\left(1 - \delta\right)}^{2} - 1}}{\textcolor{c \mathmr{and} a l}{1 - \delta - 1}}$
$\quad \quad = \frac{\textcolor{\mathrm{de} e p s k y b l u e}{1 - 2 \delta + {\delta}^{2} - 1}}{\textcolor{c \mathmr{and} a l}{- \delta}}$
$\quad \quad = \frac{\textcolor{\mathrm{de} e p s k y b l u e}{\left(2 - \delta\right) \left(- \delta\right)}}{\textcolor{c \mathmr{and} a l}{- \delta}}$
$\quad \quad = \textcolor{\mathrm{de} e p s k y b l u e}{2 - \delta}$
$\quad \quad = 2 \quad \quad \quad \quad$ as $\delta \cong 0$

limit tending

The terminology used to convey $\delta$ is close to $0$, but not $0$ is: limit delta tending to $0$.

Mathematicians use the word 'limit' to convey the meaning. We can understand this as delta is limited to very close to $0$, and it is limited such that it is not exactly $0$.

left-hand and right-hand limits

Given that function $f \left(x\right)$ evaluates to indeterminate value at $x = a$. To evaluate the possible value of $f \left(x\right) {|}_{x = a}$, The function is examined

•  at right-hand-side $x = a + \delta$ given as ${\lim}_{x \to a +} f \left(x\right)$

•  at left-hand-side $x = a - \delta$ given as ${\lim}_{x \to a -} f \left(x\right)$

Note: ${\lim}_{x \to a +}$ is read as 'limit x tending to $a$ plus'.

When a function evaluates to indeterminate value, expected values or limits at that input value is calculated for values very close to that input value.
Two limits are defined left-hand-limit and right-hand-limit.

summary

Right-hand-Limit and Left-hand-limit: Given function $f \left(x\right)$ the expected values of the function at $x = a$ are
Right-hand-limit $\textcolor{\mathrm{de} e p s k y b l u e}{{\lim}_{x \to a +} f \left(x\right) = {\lim}_{\delta \to 0} f \left(a + \delta\right)}$ and
Left-hand-limit $\textcolor{c \mathmr{and} a l}{{\lim}_{x \to a -} f \left(x\right) = {\lim}_{\delta \to 0} f \left(a - \delta\right)}$.

Outline