Section 10.1 Introduction to Probability
Subsection 10.1.1 A Very Brief Review of Probability
Recall that probability is the extent to which an event is likely to occur. Said differently, it is the ratio of the number of favourable outcomes to all possible outcomes.
Example 10.1.1. Flipping a Fair Coin.
Flipping a fair coin has two possible outcomes: Heads or Tails.
The probability of obtaining heads is 1/2, and we write \(P\)(heads) = \(\frac{1}{2}\text{.}\)
Example 10.1.2. Rolling a Fair Die.
Rolling a fair die has six possible outcomes: 1, 2, 3, 4, 5, or 6.
The probability of rolling 4 is 1/6, and we write \(P(4) = \frac{1}{6}\text{.}\)
For any event \(A\text{,}\) we have \(0 \le P(A) \le 1\text{.}\)
- If \(P(A) = 0\text{,}\) it is impossible for event \(A\) to occur.
- If \(P(A) = 1\text{,}\) event \(A\) will occur with certainty.
If \(A^c\) is the complement of \(A\text{,}\) then \(P(A^c) = 1 - P(A)\text{.}\)
Example 10.1.3.
Subsection 10.1.2 Random Variables: Discrete versus Continuous
Outcomes of random experiments, such a rolling a die or measuring the cholesterol level of a randomly chosen individual, usually are real numbers. We describe such outcomes by random variables.
A random variable \(X\) is a variable whose value may change with a random experiment. We write it as a capital letter to distinguish it from the usual variables that we have been using throughout the course.
If a random variable can take on only a finite number of values, we say it is a discrete random variable.
Example 10.1.4. A Discrete Random Variable.
If a random variable can take on any value (in an interval of real numbers), we say it is a continuous random variable.
Example 10.1.5. A Continuous Random Variable.
From now on, we concern ourselves only with continuous random variables.
Subsection 10.1.3 Probability Density Functions (PDFs)
Any continuous random variable \(X\) has a probability density function \(f\) that satisfies the following three conditions:
- \(f(x) \ge 0\text{.}\)
- \(\displaystyle \displaystyle \int_{-\infty}^{\infty} f(x) \ dx = 1.\)
- \(\displaystyle \displaystyle P(a \le X \le b) = \int_a^b f(x) \ dx.\)
Conditions 1 and 2 are needed because we are working with probabilities. Condition 3 implies that the probability that \(X\) takes on a value between \(a\) and \(b\) is the area under the probability density function between \(a\) and \(b\text{.}\)
Example 10.1.6.
For many continuous random variables, the probability density function is nonzero only on some interval of the real numbers. For example, suppose that we are analyzing a database of soil samples where the concentration of nitrogen (in g/kg) is known not to exceed 0.5 g/kg (and cannot be smaller than 0 g/kg). The function
thus is a reasonable candidate to be the probability density function for the concentration of nitrogen (in g/kg) in a randomly selected soil sample from the database.
In the following video, we work through this specific example to illustrate how to work with a given probability density function for a continuous random variable \(X\text{.}\)
Subsection 10.1.4 Cumulative Distribution Functions (CDFs)
The cumulative distribution function for the continuous random variable \(X\) is the function \(F(x)\) defined by $$F(x) = P( X \le x ).$$
Geometrically, \(F(x)\) is the area under the graph of the probability density function to the left of the value \(X = x\text{,}\) namely $$F(x) = \int_{-\infty}^x f(s) \ ds.$$
In the following video, we continue the example of the nitrogen concentration in soil samples to illustrate the relationship between the probability density function and the cumulative distribution function associated with a continuous random variable.
Subsection 10.1.5 Summary
- A random variable \(X\) is a variable whose value may change with a random experiment. A continuous random variable can take on any value (in an interval of real numbers).
-
Any continuous random variable \(X\) has a probability density function \(f\) that satisfies the following three conditions:
- \(f(x) \ge 0\text{.}\)
- \(\displaystyle \displaystyle \int_{-\infty}^{\infty} f(x) \ dx = 1.\)
- \(\displaystyle \displaystyle P(a \le X \le b) = \int_a^b f(x) \ dx.\)
- The cumulative distribution function for the continuous random variable \(X\) is the function \(F(x)\) defined by $$F(x) = P( X \le x ) = \int_{-\infty}^x f(s) \ ds.$$
Subsection 10.1.6 Don't Forget
Don't forget to return to eClass to complete the pre-class quiz.
Subsection 10.1.7 Further Study
Remember that the notes presented above only serve as an introduction to the topic. Further study of the topic will be required. This includes working through the pre-class quizzes, reviewing the lecture notes, and diligently working through the homework problems.
As you study, you should reflect on the following learning outcomes, and critically assess where you are on the path to achieving these learning outcomes:
Learning Outcomes
- Recall the definition of a continuous random variable and explain its meaning.
- Apply the definition of a probability density (PDF) function to verify whether or not a given function is a probability density function.
- Compute probabilities from a given probability density function (PDF).
- Determine the cumulative distribution function (CDF) from a given probability density function (PDF).
- Compute probabilities from a given cumulative distribution function (CDF).
- Construct a probability density function (PDF) by normalizing a given function.
- Construct and use the probability density function (PDF) for a uniform random variable on the interval \([a,b]\text{.}\)
The following references provide a good start for review and further study:
| Learning Outcome | Video | Textbook Section |
|---|---|---|
| 1 | N/A | 12.5 |
| 2 | 10.E1 | 12.5 |
| 3 | 10.E1 | 12.5 |
| 4 | 10.E2 | 12.5 |
| 5 | 10.E2 | 12.5 |
| 6 | N/A | N/A |
| 7 | N/A | N/A |