To find the expected value of the random variable exp(X), we need to calculate the integral of exp(x) multiplied by the probability density function (PDF) of X, and then evaluate it over the appropriate range.
Given that X is an exponential random variable with PDF fX(x) = 2 exp(-2x) for x ≥ 0, we want to find E[exp(X)], which is the expected value of exp(X).
The expected value of a continuous random variable can be computed using the following formula:
E[g(X)] = ∫ g(x) * fX(x) dx
In our case, we want to find E[exp(X)], so we need to compute the following integral:
E[exp(X)] = ∫ exp(x) * 2 exp(-2x) dx
Simplifying the expression:
E[exp(X)] = 2 ∫ exp(-x) dx
Now, we can integrate the expression:
E[exp(X)] = -2 exp(-x) + C
To evaluate the integral, we need to determine the limits of integration. Since X is an exponential random variable defined for x ≥ 0, the limits of integration will be from 0 to infinity.
E[exp(X)] = -2 exp(-x) |_0^∞
E[exp(X)] = -2 [exp(-∞) - exp(0)]
Since exp(-∞) approaches 0, and exp(0) = 1, we can simplify further:
E[exp(X)] = -2 [0 - 1] = 2
Therefore, the expected value of the random variable exp(X) is 2.
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I need help with this pls.
The number of sessions will be 5 and the cost is $249.
What is an expression?The mathematical expression combines numerical variables and operations denoted by addition, subtraction, multiplication, and division signs.
Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also denote the logical syntax's operation order and other properties.
Given that for the first plan, the initial cost is $39 and the cost per session is $42 and for the second plan, initial cost is $54 and per session is $39.
The number of sessions for same cost of two plans will be calculated as,
39 + 42S = 54 + 39S
42S - 39S = 54 - 39
3S = 15
S = 5
For 5 sessions the cost will be calculated as,
Cost = 39 + 42S
Cost = 39 + ( 42 x 5 )
ost = 39 + 210
Cost = $249
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What is the perimeter of a rectangle with a length of Eight and one-sixth feet and a width of Six and two-thirds feet?
- 14 3/9 ft
- 16 2/6 ft
- 29 4/6 ft
- 48 3/18 ft
The required perimeter of the given rectangle is (C) 29 4/6 ft.
What is a rectangle?A rectangle is a quadrilateral with four right angles in the Euclidean plane.
It can alternatively be described as a parallelogram with a right angle or an equiangular quadrilateral, where equiangular denotes that all of its angles are equal.
A square is a rectangle with four equally long sides.
So, the perimeter formula is:
2(l+b)
Now, calculate the perimeter as follows:
2(l+b)
2(8 1/6+ 6 2/3)
2(49/6 + 20/3)
2(89/6)
178/6
29 4/6 ft
Therefore, the required perimeter of the given rectangle is (C) 29 4/6 ft.
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Correct question:
What is the perimeter of a rectangle with a length of Eight and one-sixth feet and a width of Six and two-thirds feet?
A. 14 3/9 ft
B. 16 2/6 ft
C. 29 4/6 ft
D. 48 3/18 ft
can someone please help I can't figure this out
Step-by-step explanation:
For the first equation to make a line plot a point at negitive 5 then go up three then go one to the right
For the second equation to make a like plot a point at positive 4 then go down three then go one to the left
Please help I’m stuck on this question.
The requried simplified value of the given expression is given as -a³/4b². Option B is correct.
What is simplification?The process in mathematics to operate and interpret the function to make the function or expression simple or more understandable is called simplifying and the process is called simplification.
Here,
Given expression,
= [-b³]⁰/[2ba⁻⁷ × -2ba⁴]
= -1 / 4b²a⁻³
= -a³/4b²
Thus, the requried simplified value of the given expression is given as -a³/4b². Option B is correct.
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Use the Product or Quotient Rule to differentiate the function. 1. y =(x^2-3x)(x^2 + 1); 2. s(t) = (sqrt t+3) (4-t^2) 3. f(x)=x^4 sin x
The derivatives of the functions are as follows: y' = 4x^3+2x-3x^2-3 , s'(t) =[tex]2(2-t^2)/(2sqrt t) - 2t(sqrt t+3)/(2sqrt t)[/tex] , f'(x) = 4x^3sin x + x^4cos x
1. y' = (2x-3)(x^2+1) + (x^2-3x)(2x) = 4x^3+2x-3x^2-3
2. s'(t) = [tex](1/(2sqrt t))(4-t^2) + (sqrt t+3)(-2t) = 2(2-t^2)/(2sqrt t) - 2t(sqrt t+3)/(2sqrt t)[/tex]
3. f'(x) = 4x^3sin x + x^4cos x
1. Begin by applying the product rule to differentiate the function. The product rule states that the derivative of a product of two functions is equal to the product of the derivatives of the two functions.
y' = (2x-3)(x^2+1) + (x^2-3x)(2x)
2. Begin by applying the product rule to differentiate the function. The product rule states that the derivative of a product of two functions is equal to the product of the derivatives of the two functions.
[tex]s'(t) = (1/(2sqrt t))(4-t^2) + (sqrt t+3)(-2t)[/tex]
3. Begin by applying the quotient rule to differentiate the function. The quotient rule states that the derivative of the quotient of two functions is equal to the quotient of the derivatives of the two functions minus the product of the two functions divided by the square of the denominator.
f'(x) = 4x^3sin x + x^4cos x
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What is the two steps in solving inequality?
Two steps in solving the inequality: simplify the algebraic expressions by combining like terms and isolating the target variable, then applying the mathematic operators.
Steps in solving the inequality:- Simplify the expression by: open the parentheses of algebraic expression, combine the terms like, and isolate the target variable.
- Use mathematic operators such as: addition, subtraction, multiplication, and division. All mathematic operations must be done on both sides. Addition and subtraction do not change the inequality. Multiplication and division with positive numbers will not change the inequality.
Multiplication and division with negative numbers will change the sign of the inequality.
Example:
Solve the inequality: 3x + 2 < 5x - 8
Step one: Subtract both sides with 5x and 2, this operation does not change the inequality sign
3x + 2 - 5x - 2 < 5x - 8 - 5x - 2
-2x < -10
Step two: Divide both sides with (-2). Since we divide with negative numbers, flip the inequality sign.
-2x/(-2) > -10/(-2)
x > 5
Hence, the solution is x> 5
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A.Direct
B.Inverse
C.neither
The given graph is of inverse function.
What is inverse function?A function that can change into another function in the opposite direction is known as an inverse function or anti function. In other words, the inverse of a function "f" will take y to x if any function "f" takes x to y. If a function is written as "f" or "F," the inverse function is written as "f-1" or "F-1."
What is the inverse formula?The original value for which a function produced its output is returned by the inverse function. Consider the inverse relationship between the functions f and g: f(g(x)) = g(f(x)) = x. The initial value is fetched by a function that is its inverse. Therefore, the inverse of f is g(y) = (y-5)/2 = x. (x).
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Problem Walkthrough: Find the area of this figure.
Use 3.14 to approximate pi.
6
6
9
To solve for the area, split the figure into simple figures as
shown below. Solve for the area of each and combine.
Triangle
[?]
Rectangle
6
+
+
D
Half-circle
=
Enter the value
that belongs in
the green box.
Total Area
Enter
The total area of the figure when divided in rectangle, triangle and half circle is 86.13 square units.
What is area?Area is the room contained by a specific shape or location. It speaks of how much room is consumed. The area and perimeter of the shape will increase in size as it gets bigger.
Given figure comprises of a rectangle, triangle, and a half circle.
The area of the figure is thus calculated by adding the value of all the figures as follows:
A = Area of triangle + Area of rectangle + Area of semi-circle
[tex]A = \frac{1}{2} (b) (h) + (l)(b) + \frac{1}{2} (\pi r^{2} )\\[/tex]
Substitute the value of length, width, height and radius:
[tex]A = \frac{1}{2} (6) (6) + (9)(6) + \frac{1}{2} (3.14 (3^{2}) )\\\\\\A = 18 + 54 + 14.13\\\\A = 86.13[/tex]
Hence, the area of the figure is 86.13 square units.
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To indirectly measure the distance across a river, Arun stands on one side of the river and uses sight-lines to a landmark on the opposite bank. Arun draws the diagram below to show the lengths and angles that he measured. Find PRPR, the distance across the river. Round your answer to the nearest foot.
The distance PR, representing the distance across the river, is given as follows:
PR = 294 ft.
What are similar triangles?Similar triangles are triangles that share these two features given as follows:
Congruent angle measures.Proportional side lengths.The two similar triangles for this problem are given as follows:
POC and PRE.
Hence the distance PR is symbolized as follows:
PR = x.
The equivalent side lengths form a proportional relationship, given as follows:
x/(x + 160) = 230/355.
Hence, applying cross multiplication, the distance PR is obtained as follows:
355x = 230(x + 160)
355x = 230x + 36800
125x = 36800
x = 36800/125
x = 294 ft.
Missing InformationThe diagram is given by the image presented at the end of the answer.
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Answer: The distance across the river is 181 ft.
Step-by-step explanation:
in photo
How do you calculate the x-intercept of a line written in Standard Form?
A. Set y = 0 and solve for x.
B. Convert to Slope-Intercept Form and use the value of b.
C. The x-intercept is the coefficient of the x-term in the equation.
D. Set x = 0 and solve for y.
The y intercept of a line written in the standard form is calculated by
D. Set x = 0 and solve for y.How to solve for the interceptA linear function consists of functions where the variables has exponents of 1. The relationship is expressed in the form.
y = mx + c
definition of variable to suit the problem
y = output variable
m = slope
x = input variable
c = y intercept
To solve for the intercept in the equation y = mx + c x is set to zero
y = mx + c
y = m(0) + c
y = 0 + c
y = c
hence the intercept is equal to c when x is zero
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Why is it called equation?
The definition of an equation in algebra is a mathematical statement that proves two mathematical expressions are equal.
what is equation?Consider the equation 3x + 5 = 14, where 3x + 5 and 14 are two expressions that are separated by the word "equal." The slope-intercept form, standard form, and point-slope form are the three primary forms of linear equations. A mathematical expression known as an equation contains the equals sign. Algebra is a significant component of many equations. When you don't know the exact number to compute in math, you use algebra.
given
The definition of an equation in algebra is a mathematical statement that proves two mathematical expressions are equal.
Consider the equation 3x + 7 = 14, where 3x + 7 and 14 are two expressions that are separated by the symbol "equal."
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help is very much needed.
Why do people say geometry is hard?
Due to the lack of spatial thinking instruction in most math classes, people say geometry is hard. People should acquire geometrical ideas easily instead.
Who named geometry?The father of geometry, Euclid was a brilliant mathematician. The terms "ge" and "metria," which both imply "measuring," are the origins of geometry. For more than 2000 years, the Greeks' method of approaching geometry has served as the foundation for geometry.
The study of various mathematical or actual-world forms, figures, and sizes is known as Geometry. We learn about various angles, transformations, and similarities between the forms in geometry. The point, line, and plane are the main building blocks of geometry.
Everyone finds it difficult to dive into the issue of proofs. Children must realize that this issue affects everyone, even the mathematically skilled. Because they have trouble seeing how 3D, and 2D, forms, angles, and lines connect from various viewpoints, most youngsters struggle with geometry issues. Even kids who are really good at solving sum problems in math might be utterly stumped by this.
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Due to the lack of spatial thinking instruction in most math classes, people say geometry is hard. People should acquire geometrical ideas easily instead.
Who named geometry?Euclid, the inventor of geometry, was a master mathematician. The words "ge" and "metria," both of which denote "measuring," are where geometry got its start. The way the Greeks approached geometry has been the basis for geometry for more than 2000 years.
Geometry is the study of different mathematical or real-world shapes, sizes, and figures. Geometry teaches us about numerous angles, transformations, and similarities among forms. The primary components of geometry are the point, line, and plane.
Everyone finds it challenging to explore the topic of proofs. Children need to understand that this problem impacts everyone, even those who are good at arithmetic. Most kids have problems with geometry because they have trouble recognizing how 3D and 2D shapes, angles, and lines link from different views. Even children who excel at math sum problems may be completely baffled by this.
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An apartment has 972 square feet of carpeting. How much is this in square meters? Use the following conversion: 1 square meter is 10.8 square feet.
(Giving Brainliest, thanks, 5 stars and 20 points for correct answer cause I do that)
(it also gave me that conversion)
Answer: 90 sqare meters
Step-by-step explanation: you have to take 972 and divide it by 10.8 then you get your answer
Given a polynomial function f(x) = 2x2 + 7x + 6 and an exponential function g(x) = 2x + 5, what key features do f(x) and g(x) have in common?
Both f(x) and g(x) increase over the interval of [–4 , ∞)
Both f(x) and g(x) have the same x-intercepts of (–2, 0) and (–1.5, 0).
Both f(x) and g(x) have the same y-intercept of (0, 6).
Both f(x) and g(x) have the same range of (–∞, 0].
The solution is Option C.
The functions f ( x ) and g ( x ) have the same y-intercept of ( 0 , 6 )
What is an Equation?Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.
It demonstrates the equality of the relationship between the expressions printed on the left and right sides.
Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.
Given data ,
Let the first function be represented as A
Let the second function be represented as B
Now , the value of A is
f ( x ) = 2x² + 7x + 6 be equation (1)
Now , the value of B is
g ( x ) = 2ˣ + 5 be equation (2)
Now , on simplifying the equation , we get
To find the y-intercepts of the function ,
Substitute the value of x = 0 in the equation , we get
f ( 0 ) = 2 ( 0 )² + 7 ( 0 ) + 6
On simplifying the equation , we get
y = 6
Now , g ( x ) = 2ˣ + 5
Substitute the value of x = 0 in the equation , we get
g ( 0 ) = 2⁰ + 5
g ( 0 ) = 1 + 5
g ( 0 ) = 6
Hence , the y intercepts of the functions f ( x ) and g ( x ) is ( 0, 6 )
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Five friends together earned $165 raking leaves. They divided the money evenly among the 5 of them. Maria earned another $10 walking a neighbor's dog. How much money did Maria earn in all?Enter your answer in the box.
Find the Equation of the line that goes through (2,7) and (-1.1)
*Graphically and Algebraically
What is the slope of 5y =- 10?
The slope of line equation 5y = -10 is m = 0.
There are three type of the equation of line:
- slope-intercept form.
The equation of line is: y = mx + b
Where:
m = slope
b = y-intercept
- point-slope form
The equation of line is: y - y₁ = m(x - x₁)
Where:
m = slope
(x₁,y₁) = a point on the line
- standard form
Ax + By + C = 0
Where A, B, C are constant
The given equation of line is: 5y = -10
Convert it first to slope-intercept form. Divide both sides by 5:
5y/5 = -10/5
y = -2
Then, compare with slope intercept form: y = mx + b
Note that y = -2 can be written as:
y = 0 ·x - 2
Hence, the slope = m = 0
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Magnolia ha a net aet of P142,234. 00 after liabilitie of P53,435. 00 at the end of the
year. What i it equity at year end?
Magnolia ha a net aet of P142,234. 00 after liabilitie of P53,435. 00 at the end of the year. Then the equity at year end become 88799
magnolia ha a net aet of P142,234. 00 after liabilitie of P53,435. 00 at the end of the
year. What i it equity at year end?
The ideal current ratio for a company is between 1.2 and 2, which indicates that it has two times as many current assets as liabilities to cover its debts.
If the current ratio is less than 1, the company's liquid assets are insufficient to cover its current liabilities.
equity = assets - liabilities
= 142234-53435
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Is the square of a sum the same as the sum of squares?
A number's square root multiplied by another number's square root does not always equal the total of these two numbers' square roots.
Why is it called sum of squares?A statistical method called the sum of squares (SS) is used in regression analysis to determine how well the data can fit the model and how dispersed the data are. Since it is determined by adding up the squared differences, the sum of squares is how it is calculated.
A sum's square is equal to the product of its summands' squares and its double products' sums of summands in twos.
So, it is not always true that the square root of a sum is equal to the sum of the square roots.
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Select all the values that are equivalent to the given expression. Express your answer in scientific notation.
(9.6 × 103) × (6.7 × 102)
The correct answer is 6.432 × 10 raised to the power of 6
Select all statements that are TRUE about the graph.
A.) The pre-image is in quadrant III.
B.) The orientation of the vertices changed.
C.) The reflection can be represented by (-x, y).
D.) The line of reflection is the x-axis.
E.) The pre-image and image are congruent
HELP ASAPP
The true statements are,
A.) The pre-image is in quadrant III.
B.) The orientation of the vertices changed.
D.) The line of reflection is the x-axis.
What is orientation of the vertices?It is determined by the order in which a figure's vertices are labeled. It refers to the sequence of points after a transformation has been applied to a geometric figure. The order may be clockwise or counterclockwise.
Here, we have,
from the given figure ,we get,
the true statements are,
A.) The pre-image is in quadrant III.
B.) The orientation of the vertices changed.
D.) The line of reflection is the x-axis.
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What is obtuse triangle Class 7?
A obtuse triangle is a triangle in which one of the angles is greater than 90°.
An obtuse triangle can be mathematically defined as a triangle with one angle greater than 90°. The formula for calculating the angles of a triangle is known as the Law of Sines. The formula states that the ratio of the length of a side of the triangle to the sine of its opposite angle is the same for all sides. To calculate the angles of an obtuse triangle, we must first find the ratio of the sides to the opposite angle. For example, if we have a triangle with sides of length a, b and c, then the ratios of the sides to their opposite angles are a/sinA, b/sinB and c/sinC. Once we have found the ratios, we can use the Law of Sines to calculate the angles of the triangle. We can then determine if any of the angles are greater than 90°, which would indicate that the triangle is obtuse.
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Ayman is saving money to buy a game. The game costs $30 , and so far he has saved five-sixths of this cost. How much money has Ayman saved?
Answer: Ayman has saved $25
Step-by-step explanation: It's basically just 5/6th's of 30, which is 25.
true or false : A fair coin is flipped 5 times.
The probability of getting more heads than tails is 1/2
The probability of getting more heads than tails when a fair coin is flipped 5 times is 1/2.
True
The probability of getting more heads than tails is the same as the probability of getting an equal number of heads and tails, because the two outcomes are equally likely. The probability of getting an equal number of heads and tails is 1/2, so the probability of getting more heads than tails is also 1/2.
The probability of getting more heads than tails when a fair coin is flipped 5 times is 1/2, which is the same as the probability of getting an equal number of heads and tails.
The probability of getting more heads than tails when a fair coin is flipped 5 times is 1/2.
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Brooklyn divides 2.84 kilograms of fruit salad equally among 4 serving bowls.she wants to know how many kilograms of fruit salad to put in each bowl.she uses the decimal grids shown
By taking a quotient between the mass of salad fruit and the number of bowls, we will see that she needs to put 0.71 kg of salad fruit on each.
How much she needs to put in each bowl?
Here we know that Brooklyn wants to divide 2.84 kilograms of fruit salad equally among 4 serving bowls.
So, to see the mass of fruit salad that she needs to put in each bowl is equal to the quotient between the total mass of frit salad and 4, then we will get:
N = 2.84kg/4
N = 0.71 kg
So she needs to put 0.71 kilograms of salad fruit on each one of the 4 bowls.
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Give PQ=24 PS=19 PR=42 TQ=10 M-PQR=106 m-QSR=49 and m-PRS=35
Considering the quadrilateral, the dimensions are as follows
QR = 19SR = 24PT = 21 SQ = 20< QRS = 74< PQS = 49< RPS = 39< PSQ = 57How to find the dimensionsThe given figure is a parallelogram, and the properties will be used in finding the required dimensions
QR = PS = 19 opposite sides of a parallelogram are equal
SR = PQ = 24 opposite sides of a parallelogram are equal
PT = PR / 2 = 42 / 2 = 21 (diagonals of a parallelogram bisects each other)
SQ = 2 * TQ = 2 * 10 = 20 (diagonals of a parallelogram bisects each other)
< QRS = < QPS and < PQR = < PSR (opposite angles of a parallelogram)
< QRS + < QPS + < PQR + < PSR = 360 (sum of angles of a quadrilateral)
2 < QRS + 2< PQR = 360
2 < QRS + 2 * 106 = 360
divide through by 2
< QRS + 106 = 180
< QRS = 180 - 106
< QRS = 74
< PQS = <QSR = 49 alternate angles
< RPS + < PRS + < PSR = 180 (angles on a triangle)
< RPS + 35 + 106 = 180
< RPS + 141 = 180
< RPS = 180 - 141
< RPS = 39
<PSQ + < QSR = < PSR adjacent angles
< PSQ + 49 = 106
< PSQ = 106 - 49
< PSQ = 57
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The art class lasts 90 minutes. The teacher explains the project for 25 minutes and allows 15 minutes for questions and discussion. He talks to each student about their project for 5 minutes.Write an expression to show how many students are in the class.
The expression for the students in the class is 5N + 25 + 15 = 90 or N = 10
How to write an expression to show how many students are in the class?
An algebraic expression is an expression that is made up of variables and constants along with algebraic operations such as addition, subtraction, multiplication, exponent, etc.
Let the number of students in class be N.
Therefore, from the information provided, the expression for the students in the class can be written as:
5N + 25 + 15 = 90
5N + 40 = 90
5N = 90 - 40
5N = 50
N = 50/5
N = 10
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Write the equation of the line represented by the following graph
Answer:
Y=-[tex]\frac{1}{2}[/tex] + 4
Step-by-step explanation:
You will be using the equation of Y= Mx+B
First you must find the Y intercept, where the line crosses the Y axis, this is at positive four, this will be your B value. then you find the slope by picking 2 easily readable points on the graph, to do this i will choose (0,4) and (2,3), then you find the change in Y and put it in a fraction over the change in x, since this is going down the slope will have a negative infront of it, so your fraction is [tex]\frac{3-4}{2-0}[/tex] which equates to [tex]\frac{-1}{2}[/tex] so your slope is -1/2 (this is M value)
so lastly you plug it all in, Y= -1/2x+4
MAKE THE ANSWER IS RIGHT!!! 100 points + Brainliest. 100 points + Brainliest Let C represents the cost in dollars and R represents the revenue in dollars. What is the break-even point? Use a table to help if necessary.
C=36x+200
R=76x
Break-even point: (,)
Answer: accidentally did that sorry
Step-by-step explanation: