/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 18 Suppose that \(a\) continuous ra... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 18

Suppose that \(a\) continuous random variable has a joint probability density function given by $$f(x, y)=x^{2}+\frac{1}{3} x y, \quad 0 \leq x \leq 1, \quad 0 \leq y \leq 2$$. Find the probability that a point \((x, y)\) is in the region bounded by \(0 \leq x \leq \frac{1}{2}, 1 \leq y \leq 2,\) by evaluating the integral $$\int_{1}^{2} \int_{0}^{1 / 2} f(x, y) d x d y$$

Short Answer

Expert verified
The probability that a point $$(x, y)$$ is in the region $$0 \leq x \leq \frac{1}{2}, 1 \leq y \leq 2$$ is \frac{5}{48}.

Step by step solution

01

Set Up the Integral

Given the joint probability density function is $$f(x, y)=x^{2}+\frac{1}{3} x y$$ and the region of interest is bounded by $$0 \leq x \leq \frac{1}{2}, \quad 1 \leq y \leq 2.$$ The integral to solve is $$\text{Evaluate:} \int_{1}^{2} \int_{0}^{1 / 2} (x^{2}+\frac{1}{3} xy) \, dx \, dy.$$
02

Integrate with Respect to x

First, integrate the function with respect to x: $$\text{Compute: } \int_{0}^{1/2} (x^2 + \frac{1}{3} xy) \, dx.$$ Evaluate the integral: $$\int_{0}^{1/2} x^2 \, dx + \int_{0}^{1/2} \frac{1}{3} xy \, dx.$$ This gives: $$[ \frac{x^3}{3} ]|_{0}^{1/2} + \frac{1}{3} y [ \frac{x^2}{2} ]|_{0}^{1/2}.$$ Plug in the limits of integration. This becomes: $$\frac{(1/2)^{3}}{3} - \frac{0^{3}}{3} + \frac{1}{3} y \left(\frac{(1/2)^{2}}{2} - \frac{0^{2}}{2}\right)$$
03

Simplify the Expression

Simplify the integrated expression: $$\frac{(1/2)^{3}}{3} = \frac{1}{24}.$$ For the second term: $$\frac{1}{3} y \frac{(1/2)^{2}}{2} = \frac{1}{3} y \frac{1}{8} = \frac{y}{24}.$$ Thus, $$\frac{(1/2)^{3}}{3} + \frac{1}{3} y \frac{(1/2)^{2}}{2} = \frac{1}{24} + \frac{y}{24}.$$
04

Integrate with Respect to y

Next, integrate with respect to y: $$\int_{1}^{2} ( \frac{1}{24} + \frac{y}{24}) dy.$$ This becomes two separate integrals: $$\frac{1}{24} \int_{1}^{2} 1 \, dy + \frac{1}{24} \int_{1}^{2} y \, dy.$$ Evaluate these integrals: $$\frac{1}{24} [y]_{1}^{2} + \frac{1}{24} \frac{y^{2}}{2} |_{1}^{2}.$$
05

Evaluate the Limits

Calculate the definite integrals: $$\frac{1}{24} (2 - 1) = \frac{1}{24}.$$ For the second part: $$\frac{1}{24} \frac{2^{2} - 1^{2}}{2} = \frac{1}{24} \frac{4 - 1}{2} = \frac{1}{24} \frac{3}{2} = \frac{1}{16}.$$ Add these two results together to get the final result.
06

Final Calculation

Add the two integral values: $$\frac{1}{24} + \frac{1}{16}.$$ Find a common denominator, which is 48: $$\frac{2}{48} + \frac{3}{48} = \frac{5}{48}.$$ The final probability is: $$\frac{5}{48}.$$

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Continuous Random Variable
Continuous random variables are those variables that can take an infinite number of values within a given range. For example, the exact height of students in a class is a continuous random variable. The probability of a specific outcome for such variables is typically expressed using a probability density function (PDF). The PDF assigns a probability to the range of outcomes and not to any single outcome.
Integration
Integration is a central concept in calculus that allows us to find areas under curves, among other applications. When dealing with joint probability density functions, we often need to integrate the function over a specific region to find the probability of events occurring within that region. This involves summing infinitely small probabilities across a continuous range. For example, integrating the given joint probability density function over the specified limits helps us find the desired probability.
Limits of Integration
The limits of integration define the range over which we integrate our function. They are crucial because they specify the region in the coordinate system where we're calculating the probability. The integral \(\textstyle\bf \int\limits_{1}^{2} \int\limits_{0}^{1/2} (x^2 + \frac{1}{3}xy) \,dx \,dy\)\ ensures we only consider the values of x between 0 and 0.5 and y between 1 and 2. Properly setting these limits is essential to accurately calculating the probabilities.
Probability Calculation
Calculating probability using a continuous random variable often involves multiple steps of integration. Starting with the joint probability density function, we first integrate over one variable while treating the other as a constant. After simplifying the first integral, we integrate again over the remaining variable. For example, in our problem, we first integrated over x to get \(\frac{1}{24} + \frac{y}{24}\), then integrated over y to get the final value, \(\frac{5}{48}\). Accurate integration and careful consideration of limits lead us to the correct probability.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find \(f_{x x}, f_{x y}, f_{y x},\) and \(f_{y y}\) (Remember, \(f_{y x}\) means to differentiate with respect to \(y\) and then with respect to \(x\).) $$f(x, y)=e^{2 x y}$$

Business: maximizing production. A computer company has the following Cobb- Douglas production function for a certain product: $$ p(x, y)=800 x^{3 / 4} y^{1 / 4} $$ where \(x\) is the labor, measured in dollars, and \(y\) is the capital, measured in dollars. Suppose that the company can make a total investment in labor and capital of \(\$ 1,000,000\) How should it allocate the investment between labor and capital in order to maximize production?

Nursing facilities. A study of Texas nursing homes found that the annual profit \(P\) (in dollars) of profit-seeking, independent nursing homes in urban locations is modeled by the function $$P(w, r, s, t)=0.007955 w^{-0.638} r^{1.038} s^{0.873} t^{2.468}$$ In this function, \(w\) is the average hourly wage of nurses and aides (in dollars), \(r\) is the occupancy rate (as a percentage), s is the total square footage of the facility, and \(t\) is the Texas Index of Level of Effort (TILE), a number between 1 and 11 that measures state Medicaid reimbursement. (Source: K.J. Knox, E. C. Blankmeyer, and J. R. Stutzman, "Relative Economic Efficiency in Texas Nursing Facilities," Journal of Economics and Finance, Vol. \(23,199-213(1999) .\) Use the preceding information for Exercises 43 and 44 A profit-seeking, independent Texas nursing home in an urban setting has nurses and aides with an average hourly wage of \(\$ 20\) an hour, a TILE of \(8,\) an occupancy rate of \(70 \%,\) and \(400,000 \mathrm{ft}^{2}\) of space. a) Estimate the nursing home's annual profit. b) Find the four partial derivatives of \(P\) c) Interpret the meaning of the partial derivatives found in part (b).

Find the relative maximum and minimum values and the saddle points. $$\begin{aligned}S(b, m)=&(m+b-72)^{2}+(2 m+b-73)^{2} \\\&+(3 m+b-75)^{2}\end{aligned}$$

General interest: predicting the world record for running the mile. Note that \(x\) represents the actual year in following table. $$\begin{array}{|lc|}\hline & \text { World Record, } y \\\\\text { Year, } x & \text { (in minutes:seconds) } \\\1875 \text { (Walter Slade) } & 4: 24.5 \\\1894 \text { (Fred Bacon) } & 4: 18.2 \\\1923 \text { (Paavo Nurmi) } & 4: 10.4 \\\1937 \text { (Sidney Wooderson) } & 4: 06.4 \\\1942 \text { (Gunder H盲gg) } & 4: 06.2 \\\1945 \text { (Gunder H盲gg) } & 4: 01.4 \\\1954 \text { (Roger Bannister) } & 3: 59.6 \\\1964 \text { (Peter Snell) } & 3: 54.1 \\\1967 \text { (Jim Ryun) } & 3: 51.1 \\\1975 \text { (John Walker) } & 3: 49.4 \\\1979 \text { (Sebastian Coe) } & 3: 49.0 \\\1980 \text { (Steve Ovett) } & 3: 48.40 \\\1985 \text { (Steve Cram) } & 3: 46.31 \\\1993 \text { (Noureddine Morceli) } & 3: 44.39 \\\\\hline\end{array}$$ a) Find the regression line, \(y=m x+b,\) that fits the data in the table. (Hint: Convert each time to decimal notation; for instance, \(\left.4: 24.5=4 \frac{24.5}{60}=4.4083 .\right)\) b) Use the regression line to predict the world record in the mile in 2010 and in 2015 c) In July \(1999,\) Hicham El Guerrouj set the current (as of December 2006 ) world record of 3: 43.13 for the mile. (Source: USA Track \(\&\) Field and infoplease.com.) How does this compare with what is predicted by the regression line?
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.