To conduct a hypothesis test comparing variances of independent samples from two populations, the test statistic will have an F-distribution.
The F-distribution is a probability distribution that describes the ratio of two independent chi-squared distributions divided by their degrees of freedom. In this case, the numerator and denominator degrees of freedom are based on the sample sizes and variances of the two populations being compared.
The null hypothesis for the F-test is that the variances of the two populations are equal, and the alternative hypothesis is that they are not equal. The F-test allows us to determine if the difference in variances is statistically significant, and if we reject the null hypothesis, we can conclude that the variances are significantly different.
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Find the 66th derivative of the function f(x) = 4sin(x)
The 66th derivative of f(x) is the same as the second derivative of f(x). Thus, we can calculate the 66th derivative as follows f''(x) = -4sin(x).
What is derivative?The derivative of a function is the rate at which the function changes with respect to its input variable. It is a fundamental concept in calculus and is used in many areas of mathematics, science, and engineering.
According to question:The derivative of the function f(x) = 4sin(x) with respect to x is:
f'(x) = 4cos(x)
Taking the derivative again, we get:
f''(x) = -4sin(x)
Taking the derivative 3 times, we get:
f'''(x) = -4cos(x)
Taking the derivative 4 times, we get:
f''''(x) = 4sin(x)
We notice that the derivative of f(x) repeats every 4 times, alternating between sin(x) and cos(x) with a sign change. Therefore, to find the 66th derivative of f(x), we can simplify the calculation by considering the remainder when 66 is divided by 4:
66 mod 4 = 2
This means that the 66th derivative of f(x) is the same as the second derivative of f(x). Thus, we can calculate the 66th derivative as follows:
f''(x) = -4sin(x)
Therefore, the 66th derivative of f(x) is:
f^(66)(x) = f''(x) = -4sin(x)
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Let V be a 3 dimensional vector space with A and B its subspace of dimension 2 and 1 respectively if A
∩
B
=
0
then A
V=A-B
B
V=A+B
C
V=AB
D
none of the above
The 3-dimensional vector space represented in the form subspace dimensions A and B is given by option B. V = A + B.
V be 3-dimensional vector space.
Subspace of dimensions of A and B are 2 and 1 respectively.
And A ∩ B = 0.
It follows that every vector in A is linearly independent of every vector in B.
This implies,
Any vector v in V can be expressed uniquely as a sum of a vector in A and a vector in B.
Let v be an arbitrary vector in V.
A has dimension 2, it has a basis of two linearly independent vectors.
Let {a1, a2} be such a basis.
B has dimension 1, it has a basis consisting of a single nonzero vector b.
Then, any vector v in V can be expressed uniquely as
v = c1a1 + c2a2 + cb,
where c1, c2, and c are scalars.
Thus,
V = A + B.
Therefore, the correct answer to represents 3 dimensional vector space V as option(B). V = A + B.
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A company has a fixed cost of $1277 each day to run their factory and a variable cost of $1.93 for each widget they produce. How many widgets can they produce for $2127?
The company can produce approximately 425 widgets for $2127.
What is cost function ?
The key concept used here is the concept of cost functions, which is an important concept in economics and business. A cost function is a mathematical function that expresses the total cost of production as a function of the level of output produced. In this case, the cost function is a linear function of the form C = a + bx, where C is the total cost, a is the fixed cost, b is the variable cost per unit, and x is the level of output.
Finding the number of widgets the company can produce given a fixed cost and a variable cost per widget :
To solve this problem, we can set up an equation that relates the total cost to the number of widgets produced.
Let x be the number of widgets produced.
The total cost C is given by:
C = fixed cost + variable cost
C = 1277 + 1.93x
We want to find the number of widgets produced for a total cost of $2127. So we can set up an equation:
2127 = 1277 + 1.93x
Subtracting 1277 from both sides gives:
850 = 1.93x
Dividing both sides by 1.93 gives:
x ≈ 439.9
Since we can't produce a fractional number of widgets, we need to round down to the nearest integer. Therefore, the company can produce approximately 425 widgets for $2127.
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Which function is represented by this graph?
A. ƒ(x) = −x² + x − 6
B. f(x) = x² – 5x +6
C. f(x) = -x² + 5x - 6
D. f(x)= x²-x+6
Answer:
Equation of B represents this graph
Find Sn for the arithmetic series where a1 = 3, an = 42, n = 14
To find Sn for an arithmetic series, you can use the following formula: Sn = (n/2) * (a1 + an).
In this case, Sn = (14/2)*(3 + 42) = 189.
To explain step-by-step:
1. Find the number of terms in the series, n = 14
2. Find the first term in the series, a1 = 3
3. Find the last term in the series, an = 42
4. Plug the values into the formula, Sn = (n/2)*(a1 + an)
5. Simplify the equation and solve, Sn = (14/2)*(3 + 42) = 189
The life spans of Mr. Short and Mr. Long were
in a ratio of 3:7. Mr. Long lived 44 years longer
than Mr. Short. How long did Mr. Long live?
(30 points)
AABC is dilated by a factor of 4 to produce AA'B'C'.
A
4
37
0
8
3
5
53°
C
What is A'C', the length of AC after the dilation? What is the measure of ZA'?
OA. A'C'= 20, m
B. A'C'= 12, mZA'= 53°
OC. A'C' = 20, mA'= 148°
O
○ D. A'C'= §, m
3'
The length of AC after dilation will be 20. Then the measure of the angle ∠A' will be 37°.
We have given that,
AABC is dilated by a factor of 4 to produce AA'B'C'.
What is dilation?Dilation is the process of increasing the size of an item without affecting its form. Depending on the scale factor, the object's size can be raised or lowered.
The triangle ΔABC is dilated by a factor of 4 to produce the triangle ΔA'B'C'.
Then the length of AC after dilation. We have
A'C' = scale factor x AC
A'C' = 4 x 5
A'C' = 20
Then the measure of the angle ∠A' will be
There is no effect of dilation on the angle.
∠A = ∠A' = 37°
Thus, the correct option is A.
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define a re for the language {w | w is at least 6 symbols long and contains at least one 0 and at least one 1}
The regular expression is (0|1)[0(0|1)1(0|1)] | (0|1)[1(0|1)0(0|1)], which matches any string that is at least 6 symbols long and contains at least one 0 and at least one 1.
One possible regular expression for the language {w | w is at least 6 symbols long and contains at least one 0 and at least one 1} is:
(0|1)[0(0|1)1(0|1)] | (0|1)[1(0|1)0(0|1)]
This regular expression matches any string that satisfies the following conditions:
The string contains at least one 0 and at least one 1.
The string is at least 6 symbols long.
The string can have any number of 0s and 1s before and after the first 0 or 1, but it must contain at least one of each before and after the first 0 or 1.
For example, this regular expression matches strings like "0101010", "1000001", "1110010", but does not match strings like "101", "11111", "0000000".
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Complete question:
Alphabet = {0,1}.
Define a regular expression for the language {w | w is at least 6 symbols long and contains at least one 0 and at least one 1}
Answer the Question attached. The person who gets it right will be given brainliest.
Answer:
Example of a function that satisfies the following conditions:
a. Domain and range are both all real numbers except 5:
f(x) =
{
x if x ≠ 5
0 if x = 5
}
b. Domain is all positive numbers greater than 1, including 1:
f(x) =
{
(x-1)^2 / (x-1) if x > 1
0 if x = 1 or x = 5
}
c. Domain is all positive numbers greater than 1, but not including 1:
f(x) =
{
(x-1)^2 / (x-1) if x > 1 and x ≠ 1
0 if x = 1 or x = 5
}
Is this a quadrilateral, parallelogram, rectangle,rhombus,square or trapezoid 
As all the sides of the closed figure are equal to each other, the quadrilateral here is a square.
What is a square?A square is a closed, two-dimensional (2D), object with four corners. With four sides and four vertices, a quadrilateral is referred to as a square. All four sides of a square are equal and parallel.
In other words, a square is a polygon or quadrilateral with four sides. An equiangular quadrilateral is a shape in which all of the angles are of equal size.
Here in the given figure, we can see a quadrilateral is given.
We can see that all the sides of the quadrilateral are given to be equal to each other.
We can conclude from the observation that the quadrilateral is a square as the sides are all equal to each other.
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In the final four digits of a license
place, the sum of the first two equals the sum of
the last two. Also, the sum of the first and last
is twice the sum of the middle two, and the first
two form a two digit number that is twice that
formed by the last two. The number does not
contain any 0's. What are the final four digits
in order?
Answer:
Let the four digits be represented by abcd, where a, b, c, and d represent the digits in the thousands, hundreds, tens, and ones places, respectively.
From the first condition, we have:
a + b = c + d
From the second condition, we have:
a + d = 2(c + b)
Simplifying the second condition, we get:
a - 2b + c - d = 0
From the third condition, we have:
10a + b = 2(10c + d)
Simplifying the third condition, we get:
5a - 2b - 5c + 2d = 0
Now we have four equations with four variables. We can use substitution and elimination to solve for the variables.
From the first equation, we have:
a = c + d - b
Substituting into the second equation, we get:
c + d - b + d = 2(c + b)
Simplifying, we get:
2d - 3b + c = 0
From the third equation, we have:
10c + d = 5a
Substituting a with c + d - b, we get:
10c + d = 5(c + d - b)
Simplifying, we get:
5c - 4d + 5b = 0
Now we have two equations with two variables (2d - 3b + c = 0 and 5c - 4d + 5b = 0). Solving for b in terms of c, we get:
b = (5d - c)/3
Substituting into the first equation, we get:
2d - (5d - c)/3 + c = 0
Simplifying, we get:
7c - 13d = 0
Thus, c = 13/7d. Since c is a digit, d must be a multiple of 7. The only possible values for d are 1, 7, and 9.
If d = 1, then c = 13/7, which is not a digit.
If d = 7, then c = 13, b = 2, and a = 18. This satisfies all the conditions, and the four digits in order are 1872.
If d = 9, then c = 18, which is not a digit.
Therefore, the final four digits are 1872.
(please mark my answer as brainliest)
please answer this question
The segments which is diameter are NG and XR. So correct option is A and C.
Describe Segments?In geometry, a segment is a part of a line that is bounded by two distinct endpoints, and it contains every point on the line that is between those endpoints. A segment can be thought of as a straight line that has been cut into two parts, with a specific length. The two endpoints of a segment are typically named using capital letters, such as A and B, and the segment itself is denoted using a line over the two letters, such as AB.
Difference
In geometry, a segment and a diameter are both related to circles, but they are different concepts.
A segment is a part of a circle that is bounded by two points on the circle and a chord connecting those two points. Essentially, a segment is a portion of the circle that is cut off by a line. A circle can have many segments, depending on how the line intersects the circle.
On the other hand, a diameter is a line segment that passes through the center of the circle and has its endpoints on the circle. A diameter is the longest chord of a circle and divides the circle into two equal halves. In other words, the diameter is the distance across the circle, passing through the center.
In summary, a segment is a portion of a circle that is cut off by a line, while a diameter is a line segment that passes through the center of the circle and has its endpoints on the circle.
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Please help determine if it's linear and if so, the rate of change.
At Fred's Supermarket cans of artichoke hearts are stacked in a triangular formation for display. Each new row has 5 cans fewer than the row beneath it. ln the display there are 13 rows and the top row contains 1 can. Find the total numbers of cans in the display?
show steps
The total number of cans in the display is 611.
What is the total number of cans in display?Let's denote the number of cans in the first row as x.
According to the problem statement, each subsequent row has 5 fewer cans than the row beneath it.
Therefore, the number of cans in the second row will be x - 5, the number of cans in the third row will be x - 10, and so on.
We are given that there are 13 rows in total, and the top row has 1 can. Therefore, we can write the following equation:
x + (x - 5) + (x - 10) + ... + (x - 60) + (x - 65) = 1 + 2 + 3 + ... + 12 + 13
Simplifying the left-hand side, we can combine like terms:
13x - (5 + 10 + 15 + ... + 60 + 65) = 91
Using the formula for the sum of an arithmetic series, we can evaluate the sum of the numbers in parentheses:
5 + 10 + 15 + ... + 60 + 65 = (13/2)(5 + 65) = 455
Substituting this value into the equation, we get:
13x - 455 = 91
Solving for x, we find that:
x = 46
Therefore, the number of cans in each row is:
46, 41, 36, ..., 1
To find the total number of cans, we can use the formula for the sum of an arithmetic series:
n/2(a + l)
where;
n is the number of terms, a is the first term, and l is the last term.In this case, n = 13, a = 46, and l = 1. Plugging in these values, we get:
13/2 x (46 + 1) = 13/2 x 47 = 611.5
Since we can't have a fraction of a can, the total number of cans in the display is 611.
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2x-y=-3 then find dy/dx
Answer:
2
Step-by-step explanation:
2x-y = -3
differentiate both sides with respect to x
(2x-y)' = (-3)'
2 - dy/dx = 0
dy/dx = 2
Triangle XYZ is a 45°- 45°- 90° triangle with right angle Z. Find the coordinates of point X in
Quadrant I for Y(-1,2) and Z(6,2).
The coordinates of point X are (7/3, 2√(2)) in Quadrant I.
What is a quadrant?In coordinate geometry, a plane is divided into four regions called quadrants. These quadrants are numbered counterclockwise starting from the upper right quadrant, which is known as the first quadrant.The four quadrants are defined by the x-axis and y-axis. The x-axis is a horizontal line that runs left and right through the origin, while the y-axis is a vertical line that runs up and down through the origin.
What is isosceles triangle?An isosceles triangle is a type of triangle in which two sides are of equal length. In an isosceles triangle, the third side is called the base, and the two equal sides are called legs. The two angles formed by the legs and the base are also equal to each other. The angle opposite the base is called the vertex angle.
In the given question,
Since Triangle XYZ is a 45°- 45°- 90° triangle, the two legs of the triangle are congruent.
Let's call the length of each leg "x".We know that point Z is the right-angle vertex of the triangle and has coordinates (6, 2).
Since the triangle is isosceles, we can find the length of the other leg using the distance formula:x² + x² = (distance between Y and Z)²²x² = (6 - (-1))²²x² = 49x² = 24.5.
Now that we know the length of each leg, we can find the coordinates of point X. Since Triangle XYZ is a 45°- 45°- 90° triangle, we know that the hypotenuse is the square root of 2 times the length of a leg.
Let's call the coordinates of point X (x, y).x² + y² = (distance between X and Y)²x² + y² = x² + (y - 2)²y = 2√(2)Now we can find the x-coordinate of point X:x^2 + (2√(2))^2 = (distance between X and Z)^2x^2 + 8 = (6 - x)^2x^2 + 8 = 36 - 12x + x^212x = 28x = 7/3.
Therefore, the coordinates of point X are (7/3, 2√(2)) in Quadrant I.
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Find the equation of a parabola with a focus at (-4, 7) and a directrix of
y = 1,
Oy-7=(x+4)²
Oy-3=(x+4)²
Oy+4= (-4)²
Oy-4=(+4)²
According to the question,the equation of the parabola is y = (x + 4)² - 6.
What is equation?An equation is a statement that equates two expressions using mathematical symbols. It is a mathematical statement that two expressions are equal in value. Equations can involve numbers, variables, and constants. Equations are used to solve real-world problems such as determining the speed of a car from the distance traveled and time elapsed.
The equation of a parabola with a focus at (-4, 7) and a directrix of y = 1 is given by:
y = (x + 4)² + 4.
This equation is derived from the standard equation of a parabola:
y = (x - h)² + k,
where (h, k) is the coordinates of the focus.
In this case, the coordinates of the focus are (-4, 7), so the equation becomes:
y = (x + 4)² + 7.
The directrix of the parabola is a line, so its equation is given by:
y = 1.
Substituting this equation into the equation of the parabola, we get:
(x + 4)² + 7 = 1
(x + 4)² = -6
y = (x + 4)² - 6.
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For the functions f(x)=−7x+3 and g(x)=3x2−4x−1, find (f⋅g)(x) and (f⋅g)(1).
Answer: (f⋅g)(1) = 10.
Step-by-step explanation:
To find (f⋅g)(x), we need to multiply the two functions f(x) and g(x) together. This can be done by multiplying each term of f(x) by each term of g(x), and then combining like terms. We get:
(f⋅g)(x) = f(x) * g(x)
= (-7x+3) * (3x^2 - 4x - 1)
= -21x^3 + 28x^2 + x - 3x^2 + 4x + 1
= -21x^3 + 25x^2 + 5x + 1
To find (f⋅g)(1), we can substitute x=1 into the expression for (f⋅g)(x):
(f⋅g)(1) = -21(1)^3 + 25(1)^2 + 5(1) + 1
= -21 + 25 + 5 + 1
= 10
Therefore, (f⋅g)(1) = 10.
Given that x + 1/2 = 5, what is 2*x^2 - 3x + 6 - 3/x +2/x^2
pls help me soon
Step 1: x + 0.5 = 5
Step 2: x = 4.5
Step 3: 2*x^2 - 3x + 6 - 3/x +2/x^2 = 2(4.5)^2 - 3(4.5) + 6 - (3/4.5) + (2/(4.5)^2)
Step 4: 2*4.5^2 - 3*4.5 + 6 - 3/4.5 + 2/(4.5)^2 = 44.25 - 13.5 + 6 - 0.666666667 + 0.044444444 = 36.04444444
$690 is invested in an account earning 2.2% interest (APR), compounded quarterly.
Write a function showing the value of the account after t years, where the annual growth rate can be found from a constant in the function. Round all coefficients in the function to four decimal places. Also, determine the percentage of growth per year (APY), to the nearest hundredth of a percent.
a) A function showing the value of the account after t years, where the annual growth rate can be found from a constant, is f(x) = 690 (1+0.0055)^4t.
b) The percentage of growth per year (APY) is 2.2%.
What is a function?A function is a mathematical expression that shows the relationship between variables.
An example of a mathematical function is an equation that shows the relationship between y and x variables.
Principal = $690
APR = 2.2%
APR per quarter = 0.0055 (2.2%/4)
Compounding = Quarterly
Investment period = t years
Let f(x) = the value of the account after t years.
Future value function, (FV) = PV × (1 + r) ^ n
Where PV = present value or investment
r = compounding rate per period
n = the investment period
Therefore, f(x) or FV = 690 (1+0.0055)^4t.
APY = 100 [(1 + Interest/Principal)(365/Days in term) - 1]
2.2% = 100 [(1 + $15.18/$690)(365/365) - 1]
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9. D 10 6 D, Z, L OL, P P, J C J, E 10 2 6 X 10 2 म P Which points are on the axes 9 units from the origi 6 N 10
The correct option is C. P, J : 9 units from this origin, points are on the axes.
Explain about the coordinates?A pair of numbers called coordinates are used to locate a point or a form in a two-dimensional plane.
The x-coordinate and indeed the y-coordinate are two numbers that define a point's location on a 2D plane. The left-to-right or horizontal direction of a point is determined by the x-coordinate, which is always succeeded by the y-coordinate in an ordered pair. The vertical, up-and-down position of a point is determined by the y-coordinate.Other coordinate types include:
the North/South and East/West map coordinates.3-dimensional coordinates; polar coordinates (distance, angle);Thus, Points P, J are 9 units from this origin which lies on the negative x axis and negative y axis respectively.
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Complete question is-
Which points are on the axes 9 units from the origin?
A.
D, Z, L
B.
L, P
C.
P, J
D.
J, E
The graph the question is attached.
What is the coefficient of the fourth term in the expansion of (x - y)^4?
Answer: the distribution to the x and the 3 makes the solution 4
Step-by-step explanation: i got you broski
Alberto believes that because all squares can be called
rectangles, then all rectangles must be called squares.
Explain why his reasoning is flawed. Use mathematical
terminology to help support your reasoning.
Alberto's statement is flawed because all squares can be called rectangles, but not vice versa
Reason why Alberto's statement is flawedAlberto's reasoning is flawed because all squares can be called rectangles, but not all rectangles are squares.
While it is true that squares meet the definition of rectangles, not all rectangles meet the definition of squares.
A square is a special type of rectangle with all sides equal in length.
Therefore, Alberto's argument violates the logical concept of implication, where the truth of one proposition (squares can be called rectangles) does not necessarily imply the truth of the converse (all rectangles must be called squares).
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PLEASE HELP EASY A fair number cube is rolled twice. Determine whether each event is more or less likely than rolling the same number both times.
Select the correct button in the table to show the likelihood of each event.
Step-by-step explanation:
a probability is always the ratio
desired cases / totally possible cases
the probably for the second roll to roll the same number as in the first roll again, is actually pretty high.
it means that we "accept" any result in the first roll. that makes this a 6/6 = 1 probability.
and then for the second roll we have the usual 1/6 probability (to roll the same number again).
so, we have
6/6 × 1/6 = 1/6 = 0.166666666...
P(even, odd) is the probability to roll first an even number (3 out of 6 = 3/6 = 1/2) and then an odd number (again 3 out of 6 = 1/2) :
1/2 × 1/2 = 1/4 = 0.25
so, this is more likely.
P(2, 5) is the probability to roll first a 2 (1 out of 6 is 1/6) and then a 5 (again 1 out of 6 is 1/6) :
1/6 × 1/6 = 1/36 = 0.027777777...
so, this is less likely.
P(odd, 1) is the probability to roll first an odd number (3 out of 6 = 1/2) and then a 1 (1 out of 6 = 1/6) :
1/2 × 1/6 = 1/12 = 0.083333333...
so, it is less likely.
Find the values of x and y. ADEF = AQRS
(5y-7) ft
D
E
123⁰
F
S
S
(2x + 2)° R
= 0, y =
Q
38 ft
P
As the two triangles are congruent to each other, using that we can get the value of x = 13 and y = 9.
What are congruent triangles?Whether two or more triangles are congruent depends on the size of the sides and angles. As a result, a triangle's three sides and three angles determine its size and shape.
Two triangles are said to be congruent if their respective side and angle pairings are both equal.
Now in the given question,
The triangles are congruent so,
ED = QR
5y -7 = 38
⇒ 5y = 38+7
⇒ y = 45/5
⇒ y = 9
Now as the sum of angles in a triangle are 180°,
∠E +∠D +∠F = 180°
⇒ ∠F = 180 - 123 - 29
⇒ ∠F = 28°
As per congruency,
(2x+2) ° = 28°
⇒ 2x = 28-2
⇒ x = 26/2
⇒ x = 13
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The complete question is:
Find the values of x and y. ADEF = AQRS
(5y-7) ft
D
E
123⁰
F
S
S
(2x + 2)° R
= 0, y =
Q
38 ft
P
draw torsional moment (tmd) and torsional displacement (tdd) diagrams. label all key ordinates. what is fmax?
The f_max is the maximum value of the torsional force that can be applied to a shaft without causing it to fail.
To draw torsional moment and displacement diagrams, you can use a process called “torque diagram” which involves solving for all external moments acting on the shaft and drawing out a free body diagram of the shaft horizontally, rotating the shaft if necessary, so that all torques act around the horizontal axis. (Refer the image)
Lined up below the free body diagram, draw a set of axes. The x-axis will represent the location (lined up with the free body diagram above), and the y-axis will represent the internal torsional moment, with positive numbers indicating an internal torsional moment vector to the right and negative numbers indicating an internal torsional moment to the left. The maximum value of torsional moment is denoted by T_max or F_max.
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Pls help me right now
One month Maya rented 5 movies and 3 video games for a total of $34. The next month she rented 2 movies and 12 video games for a total of $73. Find the rental cost for each movie and each video game. Rental cost for each movie: s Rental cost for each video game: s 3 Es
The rental cost for each movie and each video game is $3.5 and $5.5 respectively.
What is the the rental cost for each movie and each video game?Let
cost of each movie = x
Cost of each video game = y
5x + 3y = 34
2x + 12y = 73
Multiply (1) by 4
20x + 12y = 136
2x + 12y = 73
subtract the equations to eliminate y
18x = 63
divide both sides by 18
x = 63/18
x = 3.5
Substitute x = 3.5 into (1)
5x + 3y = 34
5(3.5) + 3y = 34
17.5 + 3y = 34
3y = 34 - 17.5
3y = 16.5
y = 16.5/3
y = 5.5
Therefore, $3.5 and $5.5 is the rental cost of each movie and video game respectively.
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he theorem to prove is: X
is a positive continuous random variable with the memoryless property, then X∼Expo(λ)
for some λ
. The proof is explained in this video, but I will type it out here as well. I would like to get some clarification on certain parts of this proof.
Proof
Let F
be the CDF of X
, and let G(x)=P(X>x)=1−F(x)
. The memoryless property says G(s+t)=G(s)G(t)
, we want to show that only the exponential will satisfy this.
Try s=t
, this gives us G(2t)=G(t)2,G(3t)=G(t)3,...,G(kt)=G(t)k
.
Similarly, from the above we see that G(t2)=G(t)t2,...,G(tk)=G(t)1k
.
Combining the two, we get G(mnt)=G(t)mn
where mn
is a rational number.
Now, if we take the limit of rational numbers, we get real numbers. Thus, G(xt)=G(t)x
for all real x>0
.
If we let t=1
, we see that G(x)=G(1)x
and this looks like the exponential. Thus, G(1)x=exlnG(1)
, and since 0
, we can let lnG(1)=−λ
.
Therefore exlnG(1)=e−λx
and only exponential can be memoryless.
So there are several parts that I am confused about:
Why do we use G(x)=1−F(x)
instead of just F(x)
?
What does the professor mean when he says that you can get real numbers by taking the limit of rational numbers. That is, how did he get from the rational numbers mn
to the real numbers x
?
In the video, he just says that G(x)=G(1)x
looks like an exponential and thus, G(x)=G(1)x=exlnG(1)
. How did he know that this is an exponential?
G(x) is defined instead of F(x) because the property of memoryless is expressed in terms of G(x). Next, professor refers that there is rational numbers in the set of real numbers, so rational number is dense. G(x) is an exponential distribution with some rate parameter λ because G(x) has the memoryless property.
The reason why the function G(x) is defined as G(x) = P(X > x) = 1 - F(x) instead of just F(x) is because the memoryless property is expressed in terms of G(x).
Specifically, the memoryless property says G(s+t) = G(s)G(t), which means that the probability of X being greater than s+t is equal to the probability of X being greater than s multiplied by the probability of X being greater than t. This property is easier to work with when expressed in terms of G(x) rather than F(x).
When the professor says that taking the limit of rational numbers gives you real numbers, he is referring to the fact that the set of rational numbers is dense in the set of real numbers. This means that between any two real numbers, there exists a rational number.
In the context of the proof, this means that if G(mn) = G(t)^mn holds for all rational numbers mn, then it also holds for all real numbers x = mn, where mn is the limit of a sequence of rational numbers.
To see why G(x) = G(1)x looks like an exponential function, we can rewrite it as G(x) = e^(ln(G(1))x). Now, suppose we define λ = -ln(G(1)). Then we have G(x) = e^(-λx), which is the probability density function of an exponential distribution with rate parameter λ.
Thus, the assumption that G(x) has the memoryless property implies that G(x) is an exponential distribution with some rate parameter λ.
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A bag has 4 blue marbles, 3 green marbles, and 5 red
marbles. You select 2 marbles one at a time without
replacement.
Determine the probability the first marble is blue and
the second marble is green Round your answer to
the hundredths place.
The probability of selecting a blue marble on the first draw and a green marble on the second draw is 0.09.
What is probability?
Probability is a measure of the likelihood of an event occurring. It is a number between 0 and 1, where 0 means the event is impossible and 1 means the event is certain to happen.
There are 12 marbles in total in the bag, so the probability of selecting a blue marble on the first draw is 4/12.
After the first marble is drawn, there are 11 marbles left in the bag, so the probability of selecting a green marble on the second draw, given that the first marble was blue and has already been removed, is 3/11.
To determine the probability of both events occurring together, we multiply the probabilities. Therefore, the probability of selecting a blue marble on the first draw and a green marble on the second draw is:
(4/12) * (3/11) = 0.0909
Rounding to the hundredth place, the probability is approximately 0.09.
Therefore, the probability of selecting a blue marble on the first draw and a green marble on the second draw is 0.09.
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