This chapter distinguishes continuous random variables, which are measured, from discrete random variables, which are counted, using examples such as the length of a phone call or a person’s SAT score. It covers continuous probability density functions in general, and studies the uniform and exponential distributions as two specific continuous models.
Measured values versus counted values
The chapter uses paired examples, such as the number of miles driven, which is counted and discrete, against the actual distance driven, which is measured and continuous, to teach students how the same underlying quantity can be treated as either a discrete or a continuous random variable depending on exactly how it is defined.
Continuous probability density functions
Students learn to recognize and understand continuous probability density functions in general, building on the idea that probability corresponds to the area under a curve rather than to the height of a bar, extending the relative-frequency reasoning already familiar from histograms into the continuous setting.
The uniform and exponential distributions
The chapter studies two named continuous distributions in turn: the uniform distribution, where every outcome in an interval is equally likely, and the exponential distribution, and asks students to recognize each one and apply it appropriately to a described situation, building toward the normal distribution introduced later in the course.
This chapter explains continuous random variables through business-relevant examples such as rates of return from an investment and other measured, rather than counted, quantities. It covers the properties of continuous probability density functions, where probability corresponds to area under a curve, and introduces the uniform distribution as the first continuous distribution studied in detail.
Measured quantities versus counted quantities
A continuous random variable takes values from a measurement rather than a count, such as a rate of return, the length of a phone call, or the time a computer chip lasts, and the chapter contrasts this with the discrete random variables covered earlier, where the field of reliability and risk analysis depends heavily on continuous measures.
Probability as area under a curve
For a continuous distribution the graph is a curve and probability is represented by the area under that curve over a range of values, extending the relative-frequency idea from histograms; because a single point has zero width, the probability of any exact value is zero, and only the area over an interval carries meaning.
The uniform distribution
The uniform distribution is introduced as the simplest continuous probability distribution, where every value in an interval is equally likely and probabilities are found directly as proportions of the interval’s width, giving a first concrete example of how the area-under-the-curve principle is applied before moving on to more complex continuous distributions.
A continuous random variable is measured, not counted, so probability is found as area under a probability density function, not at single points. This chapter defines continuous probability functions, then develops two models: the uniform distribution, where outcomes are equally likely, and the exponential distribution, used for waiting times and decay.
Probability as Area Under a Curve
A continuous random variable is measured rather than counted, so its values fall across an interval instead of at separate points. Its behaviour is described by a probability density function, written f(x), whose graph is a curve. Probability is represented by the area between that curve and the x-axis, and the entire area under the curve equals one. Because probability is area, it is defined for intervals of x rather than for any single value, so the probability that x equals one exact point is zero.
The Uniform Distribution
In a uniform distribution every outcome in the range is equally likely, and the density function is a horizontal line of constant height. For a variable spread evenly between a and b, f(x) equals 1/(b − a) across that interval, and probabilities are found as the area of a rectangle: base times height. For example, with f(x) = 1/20 on the interval 0 to 20, the probability that x lies between 0 and 2 is (2 − 0)(1/20) = 0.1. The distribution is written X ~ U(a, b).
The Exponential Distribution
The exponential distribution models the time between events, such as how long a phone call lasts or a car battery survives, and is written X ~ Exp(m), where m is the decay parameter. Its density function is f(x) = m·e^(−m·x), and the cumulative distribution function P(X < x) = 1 − e^(−m·x) gives probability as area up to a point. Unlike the uniform distribution, smaller values are more likely than larger ones, so the curve falls away from left to right. Worked examples in the chapter use it to find warranty periods and expected lifetimes.
