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7 Additional Topics in Integration Integration by Parts
Integration Using Tables of Integrals Numerical Integration Improper Integrals Applications of Calculus to Probability
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7.1 Integration by Parts
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The Method of Integration by Parts
Integration by parts formula
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Example Evaluate Solution Let u = x and dv = ex dx
So that du = dx and v = ex Therefore, Example 1, page 484
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Guidelines for Integration by Parts
Choose u an dv so that du is simpler than u. dv is easy to integrate.
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Example Evaluate Solution Let u = ln x and dv = x dx So that and
Therefore, Example 2, page 485
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Example Evaluate Solution Let u = xex and So that and Therefore,
Example 3, page 486
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Example Evaluate Solution Let u = x2 and dv = ex dx
So that du = 2xdx and v = ex Therefore, (From first example) Example 4, page 486
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Applied Example: Oil Production
The estimated rate at which oil will be produced from an oil well t years after production has begun is given by thousand barrels per year. Find an expression that describes the total production of oil at the end of year t. Applied Example 5, page 487
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Applied Example: Oil Production
Solution Let T(t) denote the total production of oil from the well at the end of year t (t 0). Then, the rate of oil production will be given by T ′(t) thousand barrels per year. Thus, So, Applied Example 5, page 487
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Applied Example: Oil Production
Solution Use integration by parts to evaluate the integral. Let and So that and Therefore, Applied Example 5, page 487
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Applied Example: Oil Production
Solution To determine the value of C, note that the total quantity of oil produced at the end of year 0 is nil, so T(0) = 0. This gives, Thus, the required production function is given by Applied Example 5, page 487
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Integration Using Tables of Integrals
7.2 Integration Using Tables of Integrals
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A Table of Integrals We have covered several techniques for finding the antiderivatives of functions. There are many more such techniques and extensive integration formulas have been developed for them. You can find a table of integrals on pages 491 and 492 of the text that include some such formulas for your benefit. We will now consider some examples that illustrate how this table can be used to evaluate an integral.
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Examples Use the table of integrals to find Solution We first rewrite
Since is of the form , with a = 3, b = 1, and u = x, we use Formula (5), obtaining Example 1, page 493
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Examples Use the table of integrals to find Solution
We first rewrite 3 as , so that has the form with and u = x. Using Formula (8), obtaining Example 2, page 493
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Examples Use the table of integrals to find Solution
We can use Formula (24), Letting n = 2, a = – ½, and u = x, we have Example 5, page 494
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Examples Use the table of integrals to find Solution We have
Using Formula (24) again, with n = 1, a = – ½, and u = x, we get Example 5, page 494
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Applied Example: Mortgage Rates
A study prepared for the National Association of realtors estimated that the mortgage rate over the next t months will be percent per year. If the prediction holds true, what will be the average mortgage rate over the 12 months? Applied Example 6, page 495
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Applied Example: Mortgage Rates
Solution The average mortgage rate over the next 12 months will be given by Applied Example 6, page 495
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Applied Example: Mortgage Rates
Solution We have Use Formula (1) to evaluate the first integral or approximately 6.99% per year. Applied Example 6, page 495
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Numerical Integration
7.3 Numerical Integration
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Approximating Definite Integrals
Sometimes, it is necessary to evaluate definite integrals based on empirical data where there is no algebraic rule defining the integrand. Other situations also arise in which an integrable function has an antiderivative that cannot be found in terms of elementary functions. Examples of these are Riemann sums provide us with a good approximation of a definite integral, but there are better techniques and formulas, called quadrature formulas, that allow a more efficient way of computing approximate values of definite integrals.
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The Trapezoidal Rule Consider the problem of finding the area under the curve of f(x) for the interval [a, b]: y R x a b
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The Trapezoidal Rule The trapezoidal rule is based on the notion of dividing the area to be evaluated into trapezoids that approximate the area under the curve: y R1 R2 R3 R4 R5 R6 x a b
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The Trapezoidal Rule The increments x used for each trapezoid are obtained by dividing the interval into n equal segments (in our example n = 6): y x R1 R2 R3 R4 R5 R6 x a b
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The Trapezoidal Rule The area of each trapezoid is calculated by multiplying its base, x , by its average height: y x f(x0) R1 f(x1) x x0 = a x1 b
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The Trapezoidal Rule The area of each trapezoid is calculated by multiplying its base, x , by its average height: y x f(x1) R2 f(x2) x a x1 x2 b
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The Trapezoidal Rule The area of each trapezoid is calculated by multiplying its base, x , by its average height: y f(x2) R3 f(x3) x a x2 x3 b
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The Trapezoidal Rule The area of each trapezoid is calculated by multiplying its base, x , by its average height: y f(x3) R4 f(x4) x a x3 x4 b
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The Trapezoidal Rule The area of each trapezoid is calculated by multiplying its base, x , by its average height: y f(x4) R5 f(x5) x a x4 x5 b
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The Trapezoidal Rule The area of each trapezoid is calculated by multiplying its base, x , by its average height: y f(x5) R6 f(x6) x a x5 b = x6
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The Trapezoidal Rule Adding the areas R1 through Rn (n = 6 in this case) of the trapezoids gives an approximation of the desired area of the region R: y R1 R2 R3 R4 R5 R6 x a b
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The Trapezoidal Rule Adding the areas R1 through Rn of the trapezoids yields the following rule: Trapezoidal Rue
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Example Approximate the value of using the trapezoidal rule with n = 10. Compare this result with the exact value of the integral. Solution Here, a = 1, b = 2, an n = 10, so and x0 = 1, x1 = 1.1, x2 = 1.2, x3 = 1.3, … , x9 = 1.9, x10 = 1.10. The trapezoidal rule yields Example 1, page 500
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Example Approximate the value of using the trapezoidal rule with n = 10. Compare this result with the exact value of the integral. Solution By computing the actual value of the integral we get Thus the trapezoidal rule with n = 10 yields a result with an error of to six decimal places. Example 1, page 500
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Applied Example: Consumers’ Surplus
The demand function for a certain brand of perfume is given by where p is the unit price in dollars and x is the quantity demanded each week, measured in ounces. Find the consumers’ surplus if the market price is set at $60 per ounce. Applied Example 2, page 500
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Applied Example: Consumers’ Surplus
Solution When p = 60, we have or x = 800 since x must be nonnegative. Next, using the consumers’ surplus formula with `p = 60 and `x = 800, we see that the consumers’ surplus is given by It is not easy to evaluate this definite integral by finding an antiderivative of the integrand. But we can, instead, use the trapezoidal rule. Applied Example 2, page 500
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Applied Example: Consumers’ Surplus
Solution We can use the trapezoidal rule with a = 0, b = 800, and n = 10. and x0 = 0, x1 = 80, x2 = 160, x3 = 240, … , x9 = 720, x10 = 800. The trapezoidal rule yields Applied Example 2, page 500
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Applied Example: Consumers’ Surplus
Solution The trapezoidal rule yields Therefore, the consumers’ surplus is approximately $22,294. Applied Example 2, page 500
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Simpson’s Rule We’ve seen that the trapezoidal rule approximates the area under the curve by adding the areas of trapezoids under the curve: y R1 R2 x x0 x1 x2
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Simpson’s Rule The Simpson’s rule improves upon the trapezoidal rule by approximating the area under the curve by the area under a parabola, rather than a straight line: y R x x0 x1 x2
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Simpson’s Rule Given any three nonlinear points there is a unique parabola that passes through the given points. We can approximate the function f(x) on [x0, x2] with a quadratic function whose graph contain these three points: y (x2, f(x2)) (x1, f(x1)) (x0, f(x0)) x x0 x1 x2
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Simpson’s Rule Simpson’s rule approximates the area under the curve of a function f(x) using a quadratic function: Simpson’s rule
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Example Find an approximation of using Simpson’s rule with n = 10.
Solution Here, a = 1, b = 2, an n = 10, so Simpson’s rule yields Example 3, page 503
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Example Find an approximation of using Simpson’s rule with n = 10.
Solution Recall that the trapezoidal rule with n = 10 yielded an approximation of , with an error of from the value of ln 2 ≈ to six decimal places. Simpson’s rule yields an approximation with an error of to six decimal places, a definite improvement over the trapezoidal rule. Example 3, page 503
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Applied Example: Cardiac Output
One method of measuring cardiac output is to inject 5 to 10 mg of a dye into a vein leading to the heart. After making its way through the lungs, the dye returns to the heart and is pumped into the aorta, where its concentration is measured at equal time intervals. Applied Example 4, page 504
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Applied Example: Cardiac Output
The graph of c(t) shows the concentration of dye in a person’s aorta, measured in 2-second intervals after 5 mg of dye have been injected: y 3.9 4.0 4 3 2 1 3.2 2.5 1.8 2.0 1.3 0.8 0.5 0.4 0.2 0.1 x Applied Example 4, page 504
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Applied Example: Cardiac Output
The person’s cardiac output, measured in liters per minute (L/min) is computed using the formula where D is the quantity of dye injected. y 3.9 4.0 4 3 2 1 3.2 2.5 1.8 2.0 1.3 0.8 0.5 0.4 0.2 0.1 x Applied Example 4, page 504
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Applied Example: Cardiac Output
Use Simpson’s rule with n = 14 to evaluate the integral and determine the person’s cardiac output. y 3.9 4.0 4 3 2 1 3.2 2.5 1.8 2.0 1.3 0.8 0.5 0.4 0.2 0.1 x Applied Example 4, page 504
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Applied Example: Cardiac Output
Solution We have a = 0, b = 28, an n = 14, and t = 2, so that t0 = 0, t1 = 2, t2 = 4, t3 = 6, … , t14 = 28. Simpson’s rule yields Applied Example 4, page 504
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Applied Example: Cardiac Output
Solution Therefore, the person’s cardiac output is or approximately 6.0 L/min. Applied Example 4, page 504
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7.4 Improper Integrals
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Improper Integrals In many applications we are concerned with integrals that have unbounded intervals of integration. These are called improper integrals. We will now discuss problems that involve improper integrals.
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Improper Integral of f over [a, )
Let f be a continuous function on the unbounded interval [a, ). Then the improper integral of f over [a, ) is defined by if the limit exists.
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Examples Evaluate if it converges. Solution
Since ln b → , as b → the limit does not exist, and we conclude that the given improper integral is divergent. Example 2, page 513
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Examples Find the area of the region R under the curve y = e–x/ for x 0. Solution The required area is shown in the diagram below: y 1 R y = e–x/2 x 1 2 3 Example 3, page 514
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Examples Find the area of the region R under the curve y = e–x/ for x 0. Solution Taking b > 0, we compute the area of the region under the curve y = e–x/2 from x = 0 to x = b, Then, the area of the region R is given by or 2 square units. Example 3, page 514
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Improper Integral of f over (– , b]
Let f be a continuous function on the unbounded interval (– , b]. Then the improper integral of f over (– , b] is defined by if the limit exists.
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Example Find the area of the region R bounded above by the x-axis, below by y = – e2x, and on the right, by the line x = 1. Solution The graph of region R is: y 1 – 1 – 3 – 7 – x R x = 1 y = e2x Example 4, page 514
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Example Find the area of the region R bounded above by the x-axis, below by y = – e2x, and on the right, by the line x = 1. Solution Taking a < 1, compute Then, the area under the required region R is given by Example 4, page 514
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Improper Integral Unbounded on Both Sides
Improper Integral of f over (– , ) Let f be a continuous function over the unbounded interval (– , ). Let c be any real number and suppose both the improper integrals are convergent. Then, the improper integral of f over (– , ) is defined by
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Examples Evaluate the improper integral
and give a geometric interpretation of the result. Solution Take the number c to be zero and evaluate first for the interval (– , 0): Example 5, page 515
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Examples Evaluate the improper integral
and give a geometric interpretation of the result. Solution Now evaluate for the interval (0, ): Example 5, page 515
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Examples Evaluate the improper integral
and give a geometric interpretation of the result. Solution Therefore, Example 5, page 515
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Examples Evaluate the improper integral
and give a geometric interpretation of the result. Solution Below is the graph of y = xe–x2, showing the regions of interest R1 and R2: y 1 – 1 R2 – 2 – 1 x 1 2 R1 Example 5, page 515
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Examples Evaluate the improper integral
and give a geometric interpretation of the result. Solution Region R1 lies below the x-axis, so its area is negative (R1 = – ½): y 1 – 1 R2 – 2 – 1 x 1 2 R1 Example 5, page 515
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Examples Evaluate the improper integral
and give a geometric interpretation of the result. Solution While the symmetrically identical region R2 lies above the x-axis, so its area is positive (R2 = ½): y 1 – 1 R2 – 2 – 1 x 1 2 R1 Example 5, page 515
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Examples Evaluate the improper integral
and give a geometric interpretation of the result. Solution Thus, adding the areas of the two regions yields zero: y 1 – 1 R2 – 2 – 1 x 1 2 R1 Example 5, page 515
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Applications of Calculus to Probability
7.5 Applications of Calculus to Probability
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Probability Density Functions
A probability density function of a random variable x in an interval I, where I may be bounded or unbounded, is a nonnegative function f having the following properties. The total area R of the region under the graph of f is equal to 1: y R x
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Probability Density Functions
A probability density function of a random variable x in an interval I, where I may be bounded or unbounded, is a nonnegative function f having the following properties. The probability that an observed value of the random variable x lies in the interval [a, b] is given by y R1 x a b
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Examples Show that the function
satisfies the nonnegativity condition of Property 1 of probability density functions. Solution Since the factors x and (x – 1) are both nonnegative, we see that f(x) 0 on [1, 4]. Next, we compute Example 1, page 522
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Examples Show that the function
satisfies the nonnegativity condition of Property 1 of probability density functions. Solution First, f(x) 0 for all values of x in [0, ). Next, we compute Example 1, page 522
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Examples Determine the value of the constant k so that the function
is a probability density function on the interval [0, 5]. Solution We compute Since this value must be equal to one, we find that Example 2, page 522
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Examples If x is a continuous random variable for the function
compute the probability that x will assume a value between x = 1 and x = 2. Solution The required probability is given by Example 2, page 522
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Examples If x is a continuous random variable for the function
compute the probability that x will assume a value between x = 1 and x = 2. Solution The graph of f showing P(1 x 2) is: y 0.6 0.4 0.2 P(1 x 2) = 7/125 x Example 2, page 522
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Applied Example: Life Span of Light Bulbs
TKK Inc. manufactures a 200-watt electric light bulb. Laboratory tests show that the life spans of these light bulbs have a distribution described by the probability density function where x denotes the life span of a light bulb. Determine the probability that a light bulb will have a life span of 500 hours or less. More than 500 hours. More than 1000 hours but less than 1500 hours. Applied Example 3, page 523
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Applied Example: Life Span of Light Bulbs
Solution a. The probability that a light bulb will have a life span of 500 hours or less is given by Applied Example 3, page 523
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Applied Example: Life Span of Light Bulbs
Solution b. The probability that a light bulb will have a life span of more than 500 hours is given by Applied Example 3, page 523
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Applied Example: Life Span of Light Bulbs
Solution c. The probability that a light bulb will have a life span of more than 1000 hours but less than 1500 hours is given by Applied Example 3, page 523
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Exponential Density Function
The example we just saw involved a function of the form f(x) = ke–kx where x 0 and k is a positive constant, with a graph: This probability function is called an exponential density function, and the random variable associated with it is said to be exponentially distributed. Such variables are used to represent the life span of electric components, the duration of telephone calls, the waiting time in a doctor’s office, etc. y k f(x) = ke–kx x
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Expected Value Expected Value of a Continuous Random Variable
Suppose the function f defined on the interval [a, b] is the probability density function associated with a continuous random variable x. Then, the expected value of x is
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Applied Example: Life Span of Light Bulbs
Show that if a continuous random variable x is exponentially distributed with the probability density function f(x) = ke–kx (0 x < ) then the expected value E(x) is equal to 1/k. Using this result and continuing with our last example, determine the average life span of the 200-watt light bulb manufactured by TKK Inc. Applied Example 4, page 525
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Applied Example: Life Span of Light Bulbs
Solution We compute Integrating by parts with u = x and dv = e–kxdx so that We have Applied Example 4, page 525
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Applied Example: Life Span of Light Bulbs
Solution We have Applied Example 4, page 525
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Applied Example: Life Span of Light Bulbs
Solution Now, by taking a sequence of values of b that approaches infinity (such as b = 10, 100, 1000, 10,000, … ) we see that, for a fixed k, Therefore, Finally, since k = 0.001, we see that the average life span of the TKK light bulbs is 1/(0.001) = 1000 hours. Applied Example 4, page 525
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Expected Value of an Exponential Density Function
If a continuous random variable x is exponentially distributed with probability density function f(x) = ke–kx (0 x < ) then the expected (average) value of x is given by
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Applied Example: Airport Traffic
On a typical Monday morning, the time between successive arrivals of planes at Jackson International Airport is an exponentially distributed random variable x with expected value of 10 minutes. Find the probability density function associated with x. What is the probability that between 6 and 8 minutes will elapse between successive arrivals of planes. What is the probability that the time between successive arrivals of planes will be more than 15 minutes? Applied Example 5, page 527
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Applied Example: Airport Traffic
Solution Since x is exponentially distributed, the associated probability density function has the form f(x) = ke–kx The expected value of x is 10, so Thus, the required probability density function is f(x) = 0.1e–0.1x Applied Example 5, page 527
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Applied Example: Airport Traffic
Solution The probability that between 6 and 8 minutes will elapse between successive arrivals is given by Applied Example 5, page 527
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Applied Example: Airport Traffic
Solution The probability that the time between successive arrivals will be more than 15 minutes is given Applied Example 5, page 527
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End of Chapter
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