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Section 3.12 Probability
Subsection 3.12.1 Recommended Tutorials
Before starting on these exercises, you should familiarize yourself with the material covered in the following tutorials:
Subsection 3.12.2 Introduction
In this activity, you will use integration to calculate probabilities of continuous random variables. A continuous random variable is a numerical variable,
\(X\text{,}\) that can take on any value within a given range or interval. For example, perhaps
\(X\) represents the temperature in a particular classroom, which may be any value within a certain range, such as 20 to 25 degrees Celsius.
Aside: Probability Notation.
The probability notation
\begin{equation*}
P(a\lt X \lt b)
\end{equation*}
is read as "the probability that \(X\) is between \(a\) and \(b\text{.}\) " Similarly,
\begin{equation*}
P(X \lt a)
\end{equation*}
is read as "the probability that \(X\) is less than \(a\text{.}\) " The calculation of these probabilities is done by integrating the probability density function over the appropriate interval.
Every continuous random variable \(X\) has a probability density function ("pdf"), \(f\text{.}\) If you want to compute the probability that the value of \(X\) lies between \(a\) and \(b\text{,}\) then you integrate \(f\) over that interval:
\begin{equation*}
P(a\lt X \lt b)=\int_{a}^{b} f(x)dx\text{.}
\end{equation*}
Similarly, assuming that the domain of \(f\) is \((-\infty,\infty)\text{,}\) you may also compute the probability that the value of \(X\) is either less than or more than some value \(a\) using improper integration:
\begin{equation*}
P(X\lt a)=\int_{-\infty}^{a} f(x)dx\text{.}
\end{equation*}
\begin{equation*}
P(X>a)=\int_{a}^{\infty} f(x)dx\text{.}
\end{equation*}
Since a probability is always a value between 0 and 1 (or 100%), a probability density function \(f\) must always satisfy the following criteria:
\begin{align}
f(x) \amp \geq 0 \tag{3.9}\\
\int_{-\infty}^{\infty} f(x)dx \amp = 1 \tag{3.10}
\end{align}
In the following exercises, you will use two very common families of probability density functions: the exponential distribution, and the normal distribution. The probability density functions for both of these are described below, along with their mean (the expected value of
\(X\) ) and standard deviation (a measurement of the spread of observed values of
\(X\) ).
Definition 3.4 . The Exponential Distribution.
The probability density function for the exponential distribution is defined as
\begin{equation}
f(x) = \begin{cases} \lambda{ e}^{-\lambda x} \amp \text{ if } x \geq 0 \\ 0 \amp \text{ if } x\lt 0, \end{cases}\tag{3.11}
\end{equation}
where \(\frac{1}{\lambda}\) is the mean and standard deviation of the probability distribution.
Figure 3.5. The exponential probability distribution function
Definition 3.6 . The Normal Distribution.
The probability density function for the normal distribution is defined as
\begin{equation}
f(x) = \frac{1}{\sigma \sqrt{2\pi}}{ e}^{-\frac{(x-\mu)^2}{2\sigma^2}}\text{,}\tag{3.12}
\end{equation}
where \(\mu\) is the mean and \(\sigma\) is the standard deviation.
Figure 3.7. The normal probability distribution function
Exercises 3.12.3 Exercises
1. Suppose that the lifetime of a certain tire is exponentially distributed with mean
\(\frac{1}{\lambda}=45,000\) miles.
(a) Use the
piecewise() command to assign the function from equation
(3.11) using
\(\lambda = \frac{1}{45000}\text{.}\)
Hint .
Examples of the
piecewise() command can be found in
Section 9.5 .
(b) Verify that this function is a valid pdf by showing that the integral in equation
(3.10) is, in fact, equal to 1.
(c) Find the probability that a given tire will last more than 40,000 miles.
(d) Find the probability that a given tire will last less than 50,000 miles.
(e) Find the probability that a given tire will last between 40,000 and 50,000 miles.
2. Suppose that the height of a male is normally distributed with mean
\(\mu= 178\) cm and standard deviation
\(\sigma= 10\) cm.
(a) Assign the function from equation
(3.12) using these values of
\(\mu\) and
\(\sigma\text{.}\)
Hint .
Make sure to use the proper exponential function (
exp() or from the palettes toolbar) as well as the proper numerical value of
\(\pi\) (
Pi or use the palettes toolbar).
(b) Verify that this function is a valid pdf by showing that the integral in equation
(3.10) is, in fact, equal to 1.
(c) Suppose you have a friend who is
\(7\) ft tall (
\(213\) cm). Find the probability that a given individual is taller.
(d) Find the probability that a given individual is shorter than
\(213\) cm.
(e) What is the probability of selecting an individual with a height of exactly
\(213\) cm?
Hint .
Use properties of definite integrals. What is the value of
\(\displaystyle\int_a^a f(x)\;dx\text{?}\)
