Introduction

Learning objectives

Properties of Continuous Probability Distributions

The graph of a continuous probability distribution is a curve. Probability is represented by the area under the curve.

The curve is called the probability density function (abbreviated as 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 (abbreviated as 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. When we use formulas to find the area in this textbook, we are using formulas that were found by using the techniques of integral calculus. However, because most students taking this course have not studied calculus, we will not be using calculus in this textbook.

There are many continuous probability distributions. When probability is modeled by use of a continuous probability distribution, the distribution used is selected to model and fit the particular situation in the best way.

In this chapter and the next, we will study the uniform distribution, the exponential distribution, and the normal distribution. The following graphs illustrate these distributions:

Glossary

uniform distribution

a continuous random variable (RV) that has equally likely outcomes over the domain, a < x < b. Notation—X ~ U(a,b).
The mean is μ = $\frac{a+b}{2}$ and the standard deviation is $\sigma =\sqrt{\frac{{\left(b-a\right)}^2}{12}}$. The probability density function is f(x) = $\frac{1}{b-a}$ for a < x < b or a ≤ x ≤ b. The cumulative distribution is P(X ≤ x) = $\frac{x-a}{b-a}$.

exponential distribution

a continuous random variable (RV) that appears when we are interested in the intervals of time between some random events, for example, the length of time between emergency arrivals at a hospital; the notation is X ~ Exp(m).
The mean is μ = $\frac{1}{m}$ and the standard deviation is σ = $\frac{1}{m}$. The probability density function is f(x) = me^−mx, x ≥ 0 and the cumulative distribution function is P(X ≤ x) = 1 − e^−mx.