5.1 Introduction to Continuous Random Variables and The Uniform Distribution
5.1 Introduction to Continuous Random Variables and The Uniform Distribution
Learning Objectives
By the end of this chapter, the student should be able to:
- Recognize and understand continuous probability density functions
- Recognize the uniform probability distribution and apply it appropriately
Continuous random variables have many applications. Baseball batting averages, IQ scores, the length of time a long distance telephone call lasts, the amount of money a person carries, the length of time a computer chip lasts, and SAT scores are just a few. The field of reliability depends on a variety of continuous random variables.
Properties of Continuous Probability Distributions
The graph of a continuous probability distribution is a curve. Probability is represented by area under the curve.
The curve is called the probability density function (PDF). We use the symbol f(x) to represent the curve. f(x) is the function that corresponds to the graph; we use the density function f(x) to draw the graph of the probability distribution.
Area under the curve is given by a different function called the cumulative distribution function (CDF). The cumulative distribution function is used to evaluate probability as area.
- The outcomes are measured, not counted.
- The entire area under the curve and above the x-axis is equal to one.
- Probability is found for intervals of x values rather than for individual x values.
- P(c < x < d) is the probability that the random variable X is in the interval between the values c and d. P(c < x < d) is the area under the curve, above the x-axis, to the right of c and the left of d.
- P(x = c) = 0 The probability that x takes on any single individual value is zero. The area below the curve, above the x-axis, and between x = c and x = c has no width, and therefore no area (area = 0). Since the probability is equal to the area, the probability is also zero.
- P(c < x < d) is the same as P(c ≤ x ≤ d) because probability is equal to area.
We will find the area that represents probability by using geometry, formulas, technology, or probability tables. In general, calculus is needed to find the area under the curve for many probability density functions however much of the work has already been done for us. The formulas to find the area in this textbook have already been found by using the techniques of integral calculus.
Some Continuous Distributions
Probability Density Functions
We begin by defining a continuous probability density function. We use the function notation f(x). In the study of probability, the functions we study are special. We define the function f(x) so that the area between it and the x-axis is equal to a probability. Since the maximum probability is one, the maximum area is also one. For continuous probability distributions, you can think about it as: PROBABILITY = AREA.
The Uniform Distribution
The (continuous) uniform distribution is fairly simple and is a great place to start to demonstrate the ideas of co