Book cover for Calculus with Applications

Calculus with Applications

Margaret L. Lial • Raymond N. Greenwell • Nathan P. Ritchey

ISBN #9781292108971

11th Edition

3,612 Questions

Group icon
224,424 Students Helped

Homework Questions

Right arrow
Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section introduces the use of calculus in establishing continuous probability models. It explains the transition from discrete probability functions to continuous probability density functions, emphasizing the importance of non-negativity and total area equaling one. The text provides methods for calculating probabilities using definite integrals, illustrates the concepts with practical examples like bank transaction times and bird nest locations, and introduces cumulative distribution functions to encapsulate probability over intervals.

Learning Objectives

1

Describe how calculus is used to model continuous probability distributions.

2

Explain the definition and properties of a probability density function (pdf).

3

Apply definite integration to compute probabilities for continuous random variables.

4

Differentiate between discrete probability functions and continuous probability models.

5

Interpret real-world scenarios (such as bank transaction times and bird nest locations) using continuous probability concepts.

Key Concepts

CONCEPT

DEFINITION

Random Variable

A variable whose possible values are outcomes of a random phenomenon. It can be discrete (countable) or continuous (any value in an interval).

Discrete Probability Function

A function that assigns probabilities to each of a countable set of outcomes such that the sum of the probabilities equals 1.

Continuous Probability Distribution

A model in which a random variable can take any value in an interval. Probabilities are assigned over intervals rather than specific values.

Probability Density Function (pdf)

A function ƒ(x) that describes the relative likelihood for a continuous random variable to occur at a given point; it satisfies ƒ(x) ≥ 0 and the integral over its range equals 1.

Cumulative Distribution Function (cdf)

A function F(x) that gives the probability that a random variable X is less than or equal to x, defined as the integral of the pdf from the lower bound up to x.

Expected Value

The mean or average outcome of a random variable, computed as the sum (or integral) of the product of each value and its corresponding probability.

Example Problems

Example 1

Decide whether the functions defined as follows are probability density functions on the indicated intervals. If not, tell why. $$f(x)=\frac{1}{9} x-\frac{1}{18} ;[2,5]$$

Example 2

Decide whether the functions defined as follows are probability density functions on the indicated intervals. If not, tell why. $$f(x)=\frac{1}{3} x-\frac{1}{6} ;[3,4]$$

Example 3

Decide whether the functions defined as follows are probability density functions on the indicated intervals. If not, tell why. $$f(x)=\frac{x^{2}}{21} ;[1,4]$$

Example 4

Decide whether the functions defined as follows are probability density functions on the indicated intervals. If not, tell why. $$f(x)=\frac{3}{98} x^{2} ;[3,5]$$

Example 5

Decide whether the functions defined as follows are probability density functions on the indicated intervals. If not, tell why. $$(x)=\frac{x^{2}}{9} ;[0,3]$$
Scroll left
Scroll right

Step-by-Step Explanations

QUESTION

Show that the function ƒ(x) = 2x e^(-x^2) for x ≥ 0 is a valid probability density function.

STEP-BY-STEP ANSWER:

Step 1: Verify non-negativity. Note that for x ≥ 0, x is non-negative and e^(-x^2) is always positive. Thus, ƒ(x) ≥ 0 for all x in the interval.
Step 2: Verify that the total area under the curve is 1. Set up the integral: ∫ from 0 to ∞ of 2x e^(-x^2) dx.
Step 3: Use substitution. Let u = x^2 so that du = 2x dx, and the limits transform from x=0 (u=0) to x=∞ (u=∞). The integral becomes ∫ from 0 to ∞ e^(-u) du.
Step 4: Evaluate the integral, which equals 1 since ∫ from 0 to ∞ e^(-u) du = 1.
Final Answer: Since both conditions are satisfied, Æ’(x) = 2x e^(-x^2) is a probability density function.

Verification of a Probability Density Function

QUESTION

Find the probability that there is a bird’s nest within 0.5 km of a given point, given the pdf ƒ(x) = 2x e^(-x^2) for x ≥ 0.

STEP-BY-STEP ANSWER:

Step 1: Identify the interval of interest. Here, we need P(0 ≤ X ≤ 0.5).
Step 2: Set up the integral for the probability: P(0 ≤ X ≤ 0.5) = ∫ from 0 to 0.5 of 2x e^(-x^2) dx.
Step 3: Use substitution as before. Let u = x^2 (thus du = 2x dx). The limits become 0 to 0.25.
Step 4: Rewrite the integral in terms of u: ∫ from 0 to 0.25 e^(-u) du.
Step 5: Evaluate the integral: This equals [ -e^(-u) ] from 0 to 0.25 = (1 - e^(-0.25)).
Final Answer: The probability that there is a bird’s nest within 0.5 km is 1 - e^(-0.25), approximately equal to 0.2212.

Calculating the Probability for an Interval (Bird’s Nest Problem)

Scroll left
Scroll right

Common Mistakes

  • Assuming that the probability of a continuous random variable taking on any specific value is nonzero.
  • Confusing discrete histograms with continuous probability curves without appropriate smoothing or interpolation.
  • Forgetting to verify both conditions (non-negativity and total integral equaling 1) when determining if a function qualifies as a pdf.
  • Errors during substitution in the integral, which can lead to incorrect probability calculations.