Book cover for Calculus and Its Applications

Calculus and Its Applications

Larry J. Goldstein, David C. Lay, David I. Schneider

ISBN #9780134437774

14th Edition

3,033 Questions

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63,186 Students Helped

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This chapter bridges probability theory and calculus by exploring both discrete and continuous random variables and utilizing calculus techniques to compute expected values and variances. The discussion extends to in-depth explorations of exponential, normal, Poisson, and geometric distributions, highlighting their applications and importance in modeling real-world phenomena. Mastery of these concepts aids in analyzing random events and making informed decisions based on probabilistic outcomes.

Learning Objectives

1

Describe the differences between discrete and continuous random variables and identify their respective probability functions.

2

Explain how calculus is applied in computing expected values and variances for various probability distributions.

3

Analyze and interpret the properties of exponential, normal, Poisson, and geometric random variables.

4

Develop problem-solving skills through step-by-step computation and applications in real-world probability scenarios.

Key Concepts

CONCEPT

DEFINITION

Discrete Random Variable

A random variable that takes on a countable number of distinct values, each associated with a specific probability (via a probability mass function).

Continuous Random Variable

A random variable that can take any value within a given interval, described by a probability density function (pdf) and requiring integration to compute probabilities.

Expected Value

The weighted average or mean of a random variable, representing the long-run average outcome. For discrete variables, it is computed as Σ[x * p(x)], and for continuous variables, as ∫ x f(x) dx.

Variance

A measure of the spread or dispersion around the expected value, calculated as Var(X) = E(X^2) - [E(X)]^2.

Exponential Distribution

A continuous probability distribution often used to model waiting times between events, characterized by its rate parameter (λ).

Normal Distribution

A continuous, symmetric probability distribution defined by its mean and standard deviation, forming the classic bell curve.

Poisson Distribution

A discrete probability distribution that models the number of events occurring in a fixed interval of time or space, given the events occur with known constant mean rate and independently of the time since the last event.

Geometric Distribution

A discrete distribution that models the number of trials needed to achieve the first success in a sequence of independent Bernoulli trials.

Example Problems

Example 1

Table 4 is the probability table for a random variable $X .$ Find $\mathrm{E}(X), \operatorname{Var}(X),$ and the standard deviation of $X$. $$\begin{aligned}&\begin{array}{lll} \text {Table 4} \\ \hline \text { Outcome } & 0 & 1 \\\text { Probability } & \frac{1}{5} & \frac{4}{5} \\\hline\end{array}\end{aligned}$$

Example 2

Find $\mathrm{E}(X), \operatorname{Var}(X),$ and the standard deviation of $X,$ where $X$ is the random variable whose probability table is given in Table 5. $$\begin{array}{llll} \text {Table 5} \\ \hline \text { Outcome } & 1 & 2 & 3 \\\text { Probability } & \frac{4}{9} & \frac{4}{9} & \frac{1}{9} \\\hline\end{array}$$

Example 3

Compute the variances of the three random variables whose probability tables are given in Table $6 .$ Relate the sizes of the variances to the spread of the values of the random variable. $$\begin{array}{llc} \text {Table 6} \\ \hline & \text { Outcome } & \text { Probability } \\\hline & 4 & .5 \\(\mathrm{a}) & 6 & .5 \\\hline & 3 & .5 \\(\mathrm{b}) & 7 & .5 \\\hline & 1 & .5 \\(\mathrm{c}) & 9 & .5 \\\hline \end{array}$$

Example 4

Compute the variances of the two random variables whose probability tables are given in Table $7 .$ Relate the sizes of the variances to the spread of the values of the random variables. $$\begin{array}{cc} \text {Table 7} \\ \hline & \text { Outcome } & \text { Probability } \\\hline & 2 & .1 \\\text { (a) } & 4 & .4 \\& 6 & .4 \\& 8 & .1 \\\hline & 2 & .3 \\\text { (b) } & 4 & .2 \\& 6 & .2 \\& 8 & .3 \\ \hline\end{array}$$

Example 5

The number of accidents per week at a busy intersection was recorded for a year. There were 11 weeks with no accidents, 26 weeks with one accident, 13 weeks with two accidents, and 2 weeks with three accidents. A week is to be selected at random and the number of accidents noted. Let $X$ be the outcome. Then, $X$ is a random variable taking on the values $0,1,2,$ and $3.$ (a) Write out a probability table for $X$. (b) Compute $E(X).$ (c) Interpret $\mathrm{E}(X).$
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Step-by-Step Explanations

QUESTION

How do you calculate the expected value of a discrete random variable?

STEP-BY-STEP ANSWER:

Step 1: Identify the possible outcomes and their respective probabilities using the probability mass function (pmf).
Step 2: Multiply each outcome by its probability (i.e., x * p(x)).
Step 3: Sum all the products: E(X) = Σ [x * p(x)].
Final Answer: The computed sum gives the expected value of the discrete random variable.

Discrete Random Variable

QUESTION

How do you determine the expected value for a continuous random variable?

STEP-BY-STEP ANSWER:

Step 1: Identify the probability density function (pdf) f(x) for the random variable.
Step 2: Set up the integral for the expected value: E(X) = ∫ x f(x) dx over the variable's range.
Step 3: Evaluate the integral to compute the expected value.
Final Answer: The result of the integration is the expected value.

Continuous Random Variable

QUESTION

How do you calculate the variance for any random variable?

STEP-BY-STEP ANSWER:

Step 1: Compute the expected value E(X) using the appropriate method (summation for discrete or integration for continuous).
Step 2: Calculate E(X^2) using the formula (i.e., sum or integrate x^2 times the probability function).
Step 3: Apply the formula: Var(X) = E(X^2) - [E(X)]^2.
Final Answer: The computed difference gives the variance of the random variable.

Variance Calculation

QUESTION

How can calculus be used to find the probability that the waiting time exceeds a certain threshold in an exponential distribution?

STEP-BY-STEP ANSWER:

Step 1: Identify the pdf of the exponential distribution, f(x) = λe^(-λx) for x ≥ 0.
Step 2: Set up the integral for the survival function: P(X > t) = ∫ from t to ∞ λe^(-λx) dx.
Step 3: Integrate to find P(X > t) = e^(-λt).
Final Answer: The probability that the waiting time exceeds t is given by e^(-λt).

Exponential Distribution Example

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Common Mistakes

  • Confusing the methods for computing expected value in discrete versus continuous settings (summation vs. integration).
  • Forgetting to square the random variable when calculating E(X^2) for variance.
  • Misinterpreting the parameters of specialized distributions like the exponential rate parameter or the mean rate in Poisson distribution.
  • Assuming symmetry in distributions where it does not exist, such as in skewed geometric or exponential distributions.