Welcome back to class! In Section 8-5, we are shifting gears from geometry and physics to one of the most practical applications of calculus: Probability. While you may be used to calculating probabilities for discrete events (like rolling a die), calculus allows us to handle continuous random variables—things that can take on any value within a range, such as the height of a person, the lifespan of a battery, or the wait time for a customer service call.

The Probability Density Function (PDF)

At the heart of this section is the Probability Density Function, often abbreviated as $f(x)$. The crucial concept here is that probability is represented by area under the curve. To find the probability that a variable $X$ falls between two values $a$ and $b$, we integrate the function:

$$ P(a \le X \le b) = \int_{a}^{b} f(x) \, dx $$

As discussed in the lecture and class notes, for a function to be a valid probability density function, it must satisfy two strict conditions:

  • Non-negativity: $f(x) \ge 0$ for all $x$ (since you cannot have a negative probability).
  • Total Probability: The total area under the curve must equal exactly 1 (representing 100% certainty that the variable exists somewhere).
$$ \int_{-\infty}^{\infty} f(x) \, dx = 1 $$

Finding the Mean (Average Value)

Just as we used integration to find the centroid of a shape in previous chapters, we can use it to find the mean (or expected value) of a probability distribution. This is essentially finding the center of mass of the probability curve. The mean, denoted by $\mu$, is calculated as:

$$ \mu = \int_{-\infty}^{\infty} x f(x) \, dx $$

Key Distributions covered in the Notes

In the attached handwritten notes and slides, we dive into two specific types of distributions that appear frequently in the real world:

1. The Exponential Distribution

This is often used to model "waiting times" or "time to failure" (like the light bulb example in the notes). The function is generally given by $f(t) = c e^{-ct}$ for $t \ge 0$. A handy shortcut we derived in class is that the mean is the reciprocal of the constant $c$:

$$\mu = \frac{1}{c}$$

2. The Normal Distribution

You likely know this as the "Bell Curve." It is used for standardizing data like IQ scores (Example 5 in the notes) or heights. The formula involves calculating the standard deviation ($\sigma$) and the mean ($\mu$).

Class Resources

I have attached the PowerPoint slides and the handwritten class notes below. The handwritten notes walk through several step-by-step examples, including:

  • Example 1: Verifying a polynomial function is a valid PDF.
  • Example 4: Calculating the probability of a customer waiting more than 5 minutes.
  • Example 5: Analyzing IQ scores using the Normal Distribution.

Please review these materials before our next session, and happy integrating!