Welcome to Chapter 7! This week, we are moving from basic probability concepts to building full statistical models. We will be answering questions like: "How many sales can I expect to make on average?" or "What is the likelihood of exactly 3 heads in 10 coin flips?" Understanding these distributions is key to making informed decisions based on data.

1. Random Variables: Discrete vs. Continuous

Everything starts with a Random Variable ($X$), which is a numerical description of the outcome of a statistical experiment. As we discussed in the lecture notes, we categorize these into two types:

  • Discrete Random Variables: These have outcomes that we can count (e.g., $0, 1, 2, 3$).
    • Examples: The number of heads in 3 coin tosses, the score of a football game, or the number of defective circuits in a batch.
  • Continuous Random Variables: These are measured on a continuous scale.
    • Examples: The time it takes to prepare a meal, the daily price of a stock, or the amount of sugar imported.

2. Discrete Probability Distributions

For a discrete variable, we can build a table listing every possible outcome and its associated probability. Remember the two Golden Rules from our notes:

  1. The sum of all probabilities must equal 1: $\sum P(x) = 1$.
  2. Each individual probability must be between 0 and 1, inclusive.

Expected Value (The Mean)

One of the most powerful tools in this chapter is the Expected Value, denoted as $E(X)$ or $\mu$. This represents the long-run average outcome if you repeated the experiment many times. It is calculated by multiplying every possible outcome by its probability and summing them up:

$$ \mu = E(X) = \sum [x \cdot P(x)] $$

We also calculate the Standard Deviation ($\sigma$) to measure the risk or variability spread around that mean. In our class example regarding investment alternatives, looking at both the Mean and Standard Deviation helped us decide between a "safer" option and a "riskier" one.

3. The Binomial Distribution

A specific and very common type of discrete distribution is the Binomial Distribution. This applies when an experiment satisfies four specific conditions (often remembered by the nature of a coin flip):

  • There are a fixed number of trials ($n$).
  • There are only two outcomes: Success or Failure.
  • The probability of success ($p$) is constant for each trial.
  • The trials are independent.

While the probability formula for Binomial distributions involves combinations (${}_nC_x$), the formulas for the center and spread are quite elegant:

  • Mean: $\mu = n \cdot p$
  • Standard Deviation: $\sigma = \sqrt{n \cdot p \cdot (1 - p)}$

4. The Continuous Uniform Distribution

Finally, we touched on the Uniform Distribution. Imagine a situation where every outcome between a minimum value ($a$) and a maximum value ($b$) is equally likely—like a delivery truck arriving anytime between 8:00 and 8:30. The geometry here is simple: it forms a rectangle.

Since the total area (probability) must be 1, the height of our rectangle is always:

$$ \text{Height} = \frac{1}{b - a} $$

To find the probability of an event, we simply calculate the area of the rectangle corresponding to that specific range.

Next Steps: Make sure to utilize the Probability Distributions Calculator and Binomial Distribution Calculator linked above to check your work on the homework problems. Mastering these concepts now is essential for our future work in hypothesis testing!