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We begin by defining a continuous probability density function. We use the function notation $f(x)$ . Intermediate algebra may have been your first formal introduction to functions. In the study of probability, the functions we study are special. We define the function $f(x)$ so that the area between it and the x-axis is equal to a probability. Since the maximum probability is one,the maximum area is also one.
For continuous probability distributions, PROBABILITY = AREA.
Consider the function $f(x)=\frac{1}{20}$ for $0\le x\le \mathrm{20}$ . $x$ = a real number. The graph of $f(x)=\frac{1}{20}$ is a horizontal line. However, since $0\le x\le \mathrm{20}$ , $f(x)$ is restricted to the portion between $x=0$ and $x=20$ , inclusive .
$f(x)=\frac{1}{20}$ for $0\le x\le 20$ .
The graph of $f(x)=\frac{1}{20}$ isa horizontal line segment when $0\le x\le \mathrm{20}$ .
The area between $f(x)=\frac{1}{20}$ where $0\le x\le \mathrm{20}$ and the x-axis is the area of a rectangle with base = $20$ and height = $\frac{1}{20}$ .
$\text{AREA}=20\cdot \frac{1}{20}=1$
This particular function, where we have restricted $x$ so that the area between the function and the x-axis is 1, is an example of a continuousprobability density function. It is used as a tool to calculate probabilities.
Suppose we want to find the area between $f(x)=\frac{1}{20}$ and the x-axis where $0< x< 2$ .
$\text{AREA}=(2-0)\cdot \frac{1}{20}=0.1$
$(2-0)=2=\text{base of a rectangle}$
$\frac{1}{20}$ = the height.
The area corresponds to a probability. The probability that $x$ is between 0 and 2 is 0.1, which can be written mathematically as $\text{P(0<x<2)}=\text{P(x<2)}=0.1$ .
Suppose we want to find the area between $f(x)=\frac{1}{20}$ and the x-axis where $4< x< 15$ .
$\text{AREA}=(15-4)\cdot \frac{1}{20}=0.55$
$(15-4)=11=\text{the base of a rectangle}$
$\frac{1}{20}$ = the height.
The area corresponds to the probability $P(4< x< 15)=0.55$ .
Suppose we want to find $P(x=\mathrm{15})$ . On an x-y graph, $x=\mathrm{15}$ is a vertical line. A vertical line has no width (or 0 width). Therefore, $P(x=\mathrm{15})=\text{(base)}\text{(height)}=\left(0\right)\left(\frac{1}{20}\right)=0$ .
$P(X\le x)$ (can be written as $P(X< x)$ for continuous distributions) is called the cumulative distribution function or $\mathrm{CDF}$ . Notice the "less than or equal to" symbol. We can use the $\mathrm{CDF}$ to calculate $P(X> x)$ . The $\mathrm{CDF}$ gives "area to the left" and $P(X> x)$ gives "area to the right." We calculate $P(X> x)$ for continuous distributions as follows: $P(X> x)=1-P(X< x)$ .
Label the graph with $\mathrm{f(x)}$ and $x$ . Scale the x and y axes with the maximum $x$ and $y$ values. $f(x)=\frac{1}{20}$ , $0\le x\le \mathrm{20}$ .
$P(2.3< x< 12.7)=\left(\text{base}\right)\left(\text{height}\right)=(12.7-2.3)\left(\frac{1}{20}\right)=0.52$
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