Consider a random variable X with a mean of μX. You draw a random sample of 6 observations: {x1,x2,x3,x4,x5,x6}. Prove that the following estimator is an unbiased estimator of muX : μX=0.25x1+0.65x2+0.15x4−0.05x6
Answers
The estimator μ X = 0.25x₁ + 0.65x₂ + 0.15x₄ - 0.05x₆ is an unbiased estimator of μ X, the mean of random variable X. We can draw a random sample of 6 observations: {x1,x2,x3,x4,x5,x6}.
To prove this, we need to show that the expected value of the estimator is equal to the true population mean, E[μ X] = μ X.
Let's calculate the expected value of the estimator:
E(μ X) = E(0.25x1 + 0.65x2 + 0.15x4 - 0.05x6)
Using the linearity of expectation, we can split the expectation across the terms:
E(μ X) = 0.25E(x1) + 0.65E(x2) + 0.15E(x4) - 0.05E(x6)
Since the sample observations are drawn from the random variable X, the expected values of the individual observations will be equal to the true mean μ X:
E(μ X) = 0.25μ X + 0.65μ X + 0.15μ X - 0.05μ X
Simplifying the expression, we get:
E(μ X) = μ X
Taking the expected value of the estimator, we have:
E[μ X] = E[0.25x₁ + 0.65x₂ + 0.15x₄ - 0.05x₆]
= 0.25E[x₁] + 0.65E[x₂] + 0.15E[x₄] - 0.05E[x₆]
Since x₁, x₂, x₄, and x₆ are observations from the random variable X, their expected values are equal to the population mean μ X. Therefore, we can substitute μ X for E[x₁], E[x₂], E[x₄], and E[x₆]:
E[μ X] = 0.25μ X + 0.65μ X + 0.15μ X - 0.05μX= μX
Hence, we have shown that the expected value of the estimator is equal to the true population mean, E[μ X] = μ X. Therefore, the estimator μ X = 0.25x₁ + 0.65x₂ + 0.15x₄ - 0.05x₆ is an unbiased estimator of μ X.
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Related Questions
The velocity of a particle. P. moving along the x-axis is given by the differentiable function v, where (t) is measured in meters per hour and r is measured in hours. V() is a continuous and decreasing function Selected values of v(f) are shown in the table above. Particle P is at the t= 30 at time t = 0. T(hours) 0 2 4 7 10 V(t) (meters/hour) 20.3 14.4 10 7.3 5 (a) Use a Right Riemann sum with the four subintervals indicated by the data in the table to approximate the displacement of the particle between 0 hr to 10 hr. What is the estimated position of particle Pat t=10? Indicate units of measure. (b) Does the approximation in part (a) overestimate or underestimate the displacement? Explain your reasoning (c) A second particle, Q. also moves along the x-axis so that its velocity for O<=T<= 10 is given by VQ(t) = 35✓t cos( 0.06t^2) meters per hour. Find the time interval during which the velocity of particle vo(t) is at least 60 meters per hour. Find the distance traveled by particle Q during the interval when the velocity of particle Q is at least 40 meters per hour. (d) At time t = 0, particle Q is at position x = -90. Using the result from part (a) and the function vo(t) from part (c), approximate the distance between particles P and Q at time t = 10.
Answers
The velocity of a particle. P. moving along the x-axis is given by the differentiable function v, where (t) is measured is given by:
A differential function v gives the velocity of a particle P travelling down the x-axis, where v(t) is measured in metres per hour and t is measured in hours. v(t) is a declining function that is continuous. The table below shows several examples of v(t) values.
T [hours] 0 2 4 7 10
v(t) [meters/hour] 20.3 14.4 10 7.3 5
a) We know that the particle's displacement is the area under the curve v(t). We can calculate the particle's displacement by integrating v(t). Because v(t) is a monotonous (constantly declining) differentiable function, it is also Riemann Integrable. There are now five non-uniform subdivisions:
Partition t0 t1 t2 t3 t4
T [hours] 0 2 4 7 10
v(t) [meters/hour] 20.3 14.4 10 7.3 5
Using Right Riemann sum to approximate the displacement of particle between 0 hr and 10 hr is given by:
\(\sum_{n=1}^{4}v(t_n)\Delta t_n=v(t_1)(t_1-t_0)+v(t_2)(t_2-t_1)+v(t_3)(t_3-t_2)+v(t_4)(t_4-t_3) \\=(14.4)(2)+(10)(2)+(7.3)(3)+(5)(3) \\=28.8+20+21.9+15 \\=85.7\)
Therefore, the total displacement between 0 hr and 10 hr is is 85.7 meters.
The estimated position of particle P at time t = 10 hour is 115.7 (= 30 +85.7) meters.
b) Because the function v(t) is decreasing and we are estimating the integral using the Right Riemann sum, the approximation in part(a) underestimates the displacement.
c) A second particle Q also moves along the x-axis so that its velocity is given by :
\(V_Q(t)=35\sqrt{t}\cos(0.06t^2)\text{ meters per hour for }0\leq t\leq 10.\)
Hence, the time interval during which the velocity of a particle is atleast 60 meters per hour is [9.404, 10].
Now, the time periods during which a particle's velocity is at least 40 metres per hour are [1.321,4.006] and [9.218, 10]. The distance travelled by the particle Q when its velocity is at least 40 metres per hour is then calculated. :
\(\int_{1.321}^{4.006}v_Q(t)dt+\int_{9.218}^{10}v_Q(t)dt\\\\=\int_{1.321}^{4.006}35\sqrt{t}\cos(0.06t^2)dt+\int_{9.218}^{10}35\sqrt{t}\cos(0.06t^2)dt\)
d) At time t = 0, particle Q is is at position x = -90.
We know that P is at xp = 115.7 meters.
Now, The position of Q at t = 10 hr is xq:
\(x_q=-90+\int_{0}^{10}v_Q(t)dt=-90+\int_{0}^{10}35\sqrt{t}\cos(0.06t^2)dt\)
And the distance between Q and P is given by :
\(|x_p-x_q|=|115.7-(-90+\int_{0}^{10}35\sqrt{t}\cos(0.06t^2)dt)|\)
\(\\=|205.7-\int_{0}^{10}35\sqrt{t}\cos(0.06t^2)dt|\)
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Emily is three years older than twice her sister Mary's age. The sum of their ages is less than 30. Let x represent Mary's age. Which inequality represents Mary's possible age?
Answers
Answer:
0 < x < 9
Step-by-step explanation:
Mary's age = x
Twice of Mary's age = 2x
Emily's age = (2x + 3) (we are told Emily is 3 years older than twice of Mary's age)
Given that the sum of their ages is > 30, we have the following expression:
\( (2x + 3) + x \) > 30
Let's simply:
\( 2x + 3 + x \) > 30
\( 3x + 3 \) > 30
Subtract 3 from both sides
\( 3x + 3 - 3 \) > \( 30 - 3 \)
\( 3x \) > \( 27 \)
Divide both sides by 3
\( \frac{3x}{3} \) > \( \frac{27}{3} \)
\( x \) > \( 9 \)
Therefore, the inequality that represents Mary's possible age is: 0 < x < 9
What is the distance between the points? Round to the nearest 10th if necessary
Answers
Answer:
5\sqrt(5) or about 11.2
Step-by-step explanation:
The change in x is 5, the change is y is 10.
Using the pythagorean theorem, the distance is
\sqrt(25+100) = \sqrt(125) = 5\sqrt(5) or 11.2
what do you call a fake diamond in Ireland
Answers
In Ireland a Sham is known as a fake diamond.
What does Fake diamond mean in Ireland?In Ireland the term fake diamond has a different meaning than it does in other countries. In Ireland the term fake diamond is used to refer to some fake object or situation. Therefore, the word Sham is associated with the term Fake diamond in Ireland.
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what option on the proc ttest statement changes the type of alternative hypothesis tested? (simply put the option with nothing else. for instance if the question asked which option is used to specify your data set, the answer would just be data.) answer: question 16
Answers
ALTERNATIVE option is used to specify your data set.
The 'alternative' option on the proc ttest statement allows the user to specify the type of alternative hypothesis they would like to test.
This option is typically set to 'two-sided' by default, which means that the user is testing to see if the mean of their data set is different from the hypothesized mean.
However,
The user can also set the alternative option to 'greater' or 'less' to test for one-sided hypotheses.
For example, if one wanted to test if the mean of their data set is greater than the hypothesized mean, they could set the alternative option to 'greater'.
A confidence interval is a range of values that is used to estimate an unknown population parameter.
The confidence interval is calculated using a sample statistic and a margin of error, which is based on the level of confidence the user has in their estimate.
Confidence intervals provide an indication of the precision of the estimate; the wider the confidence interval, the less precise the estimate is.
For example, a 95% confidence interval could be used to indicate that the true population parameter is 95% likely to fall within the range of the confidence interval.
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write a polynomial that has zeros of -3,2 and 4
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\(\begin{cases} x = -3 &\implies x +3=0\\ x = 2 &\implies x -2=0\\ x = 4 &\implies x -4=0\\ \end{cases} \\\\[-0.35em] ~\dotfill\\\\ \stackrel{original~polynomial}{ ( x +3 )( x -2 )( x -4 ) = \stackrel{0}{y}}\implies (x+3)(x^2-6x+8)=y \\\\\\ x^3-6x^2+8x+3x^2-18x+24=y\implies \boxed{x^3-3x^2-10x+24=y}\)
Find the equation to the tangent line to the curve at the specified point:-F(x) = - 2/3(x^3) + x^2 + 6x – 4at the point: (3, 5)
Answers
Given:
\(\begin{gathered} f(x)=-\frac{2}{3}x^3+x^2+6x-4 \\ \text{ Point = (3,5)} \end{gathered}\)The general equation of tangent line is:
\(y-y_1=m(x-x_1)\)Where,
\(\begin{gathered} (x_1,y_1)=\text{ Point} \\ m=\text{ Slope} \end{gathered}\)Solpe of line is:
\(\begin{gathered} m=f^{\prime}(x) \\ \end{gathered}\)\(\begin{gathered} f(x)=-\frac{2}{3}x^3+x^2+6x-4 \\ f^{\prime}(x)=-\frac{2}{3}(3x^2)+2x+6 \\ x=3;y=5 \\ f^{\prime}(x)=-2x^2+2x+6 \\ f^{\prime}(3)=-2(3)^2+2(3)+6 \\ f^{\prime}(3)=-18+6+6 \\ f^{\prime}(3)=-6 \end{gathered}\)slope of tangent line is -6
then equation of tengent line is:
\(\begin{gathered} y-y_1=m(x-x_1) \\ m=-6 \\ (x_1,y_1)=(3,5) \\ so, \\ y-5=-6(x-3) \\ y-5=-6x+18 \\ y+6x=18+5 \\ y+6x=23 \end{gathered}\)
Which equation is not true?
1). 4 ( z - 11 ) = 4z - 44
2). -2 ( 3x + 5 ) = -6x + 10
3). 0.5 ( 2m - 3) = 1m - 1,5
Answers
Answer:
2) incorrect, should be -6x - 10
Step-by-step explanation:
Decide whether the random variable x is discrete or continuous. X represents the number of motorcycle accidents in one year in California. Is the random variable x discrete or continuous? Choose the correct answer below. A. Discrete B. Continuous
Answers
A discrete random variable, x, represents the total number of motorbike accidents that occurred in California for a given year. Correct option is A.
A discrete random variable has gaps between each of its countable possible values. Since motorcycle accidents are discrete occurrences that can be counted, the total number of motorcycle accidents in this situation can only be a non-negative whole integer (0, 1, 2, 3,...).
On the other hand, continuous random variables can have an endless number of values inside a range or interval. They often reflect quantities like height, weight, or time, which can all fall within a certain range of values.
Motorcycle accidents have a variable x that can only take on whole numbers (e.g., 1.5 accidents), thus fractional or non-integer values for the number of accidents are meaningless. The random variable x is therefore regarded as discrete.
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suppose that a large mixing tank initially holds 400 gallons of water in which 80 pounds of salt have been dissolved. pure water is pumped into the tank at a rate of 4 gal/min, and when the solution is well stirred, it is then pumped out at the same rate. determine a differential equation for the amount of salt a(t) in the tank at time
Answers
The differential equation for the amount of salt A(t) in the tank at the time (dA / dT) / (A / 100) = 0, A(0) = 30
It is Given that a large tank holds 400 gallons of water, in which 80 ponds of salt have been dissolved, given that pure water is pumped into the tank at a rate of 8 gallons/min. here we have to find the differential equation for the amount of salt in the tank at the time when the solution is well stirred. we have to multiply both 400 gallons of water and pumped water at the rate of 8 gallons/min.
dA / dT = Rin - Rout
Rin = 0
Rout = A(t) / 400 * 8
= A / 50
So,
dA / dt = Rin - Rout
dA / dt = 0 - A /50
After solving the differential equation we get
A(0) = 30
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Ben's family was driving 945 miles to visit relatives. The family van averages 24 miles per gallon, and the tank holds 18 gallons of gas.
About how many times will they have to fill up the tank to arrive at their destination?
Question 21 options:
1 time
39 times
3 times
5 times
Answers
The family averages 24 miles per gallon, hence they will have to refill 3 times.
An average is simply a single data point chosen to represent a series of data, typically the sum of the elements divided by the sum of the numbers in the collection. For example, the average of the integers 2, 3, 4, 7, and 9 is 5.
A rate in mathematics is the ratio of two related variables stated in different units. If one of these variables is given as a single unit in the denominator of the ratio, and it is assumed that this quantity can be changed on a regular basis (i.e., is a variable and the independent), the numerator of the ratio expresses the corresponding rate of change in another (dependent) variable.
Total ride is one tank of gas = 18 × 24 = 432 miles.
Total distance = 945 ÷ 432 = 2.18 ≈ 3 (taking the greater value)
Hence it will be required to fill the tank 3 times to arrive at the destination.
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find the volume of the given solid. bounded by the coordinate planes and the plane 6x + 4y + z = 24
Answers
Therefore, the volume of the solid bounded by the coordinate planes and the plane 6x + 4y + z = 24 is 96 cubic units.
To find the volume of the solid bounded by the coordinate planes (xy-plane, xz-plane, and yz-plane) and the plane 6x + 4y + z = 24, we need to determine the region in space enclosed by these boundaries.
First, let's consider the plane equation 6x + 4y + z = 24. To find the x-intercept, we set y = 0 and z = 0:
6x + 4(0) + 0 = 24
6x = 24
x = 4
So, the plane intersects the x-axis at (4, 0, 0).
Similarly, to find the y-intercept, we set x = 0 and z = 0:
6(0) + 4y + 0 = 24
4y = 24
y = 6
So, the plane intersects the y-axis at (0, 6, 0).
To find the z-intercept, we set x = 0 and y = 0:
6(0) + 4(0) + z = 24
z = 24
So, the plane intersects the z-axis at (0, 0, 24).
We can visualize that the solid bounded by the coordinate planes and the plane 6x + 4y + z = 24 is a tetrahedron with vertices at (4, 0, 0), (0, 6, 0), (0, 0, 24), and the origin (0, 0, 0).
To find the volume of this tetrahedron, we can use the formula:
Volume = (1/3) * base area * height
The base of the tetrahedron is a right triangle with sides of length 4 and 6. The area of this triangle is (1/2) * base * height = (1/2) * 4 * 6 = 12.
The height of the tetrahedron is the z-coordinate of the vertex (0, 0, 24), which is 24.
Plugging these values into the volume formula:
Volume = (1/3) * 12 * 24
= 96 cubic units
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Which polynomial is factored completely? 4 (4 x superscript 4 baseline minus 1) 2 x (y cubed minus 4 y squared 5y) 3 x (9 x squared 1) 5 x squared minus 17 x 14
Answers
The answer choice which represents a polynomial which is factored completely is; (y³ - 4y² +5)
Which polynomial is factored completely?The polynomials as given in the task content are;
4(4x⁴ -1) + 2x
(y³ - 4y² +5)
3x (9x² -1)
5x² -17x +14
Hence it follows that the polynomial which cannot be further factorised is; (y³ - 4y² +5).
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Answer:3 x (9 x squared + 1)
it is c because i did the test and put b but got it wrong
Step-by-step explanation:
if 3/5 x +10 =17, then x=?
Answers
Answer:
x = 11.6 or x = 11 2/3 or x = 35/3
Step-by-step explanation:
\(\qquad\qquad\huge\underline{{\sf Answer}}\)
let's evaluate the value of x ~
\(\qquad \sf \dashrightarrow \: \dfrac{3}{5} x + 10 = 17\)
\(\qquad \sf \dashrightarrow \: \dfrac{3}{5} x = 17 - 10\)
\(\qquad \sf \dashrightarrow \: \dfrac{3}{5} x = 7\)
\(\qquad \sf \dashrightarrow \:x = 7 \cdot\dfrac{5}{3} \)
\(\qquad \sf \dashrightarrow \:x = \dfrac{35}{3} \)
or
\(\qquad \sf \dashrightarrow \:x \approx11.66\)
3x^2-5x+5y-4+12x
WILL GIVE BRAINLY
Answers
Find the value of P.
Answers
Answer:
p = 10
Step-by-step explanation:
The dashes on the lines show they are equal in length.
Therefore ΔRUL ≅ ΔAUL
and m∠RLU = m∠AUL
⇒ 5p - 8 = 2p + 22
⇒ 3p - 8 = 22
⇒ 3p = 30
⇒ p = 30÷ 3
⇒ p = 10
(Please Help) Use your knowledge of the Triangle Sum Theory to solve for x. Then solve for L.
x = ?
K =
Answers
Answer:
X= 15
K=45
Step-by-step explanation:
The Triangle Sum Theory means that every interior angle of a triangle sums up to 180. My first approach was to put 169 as x but that isn’t correct. That is because a triangle’s smallest angles have to add up to be more than the biggest angle. Ex (10, 5, 7 that is a triangle because the sum of 5+7 is more than 10) So as you can see there are x’s next to the other numbers. That is because x needs to be able to multiply with the other numbers and stand on it’s own and add up to 180. So this would be the equation: X+8•X+3•X=180. You should find that 15 makes everything add up to 180. Now knowing x is 15 now we know that 3x is equal to 45. So k=45 That is how you solve this tricky question. Hope this helps! ❤️
Find the missing length of the triangle.
C = 30
B = 24
A = ?
Answers
Answer:
A = 18
Step-by-step explanation:
A = √30^2-24^2
A = 18
Arya deals in clothing and accessories. She bought 10 shirts at a wholesale price of 1120 each and sold them
all at a profit of 40 percent.
What was the price Arya charged for the 10 shirts?
Answers
Answer:
15680
Step-by-step explanation:
Cost of 1 shirt: 1120
Cost of 10 shirts: 11200
Add 40% to the cost of 10 shirts:
40% + 100% = 140% = 1.4
1.4 * $11200 = 15680
Answer: 15680
2/5 of Ashley's fruit are strawberries 1/4. of the strawberries are chocolate covered what fraction of Ashley's fruit covered strawberries? (your answer should be in simplest form)
Answers
We know that
• 2/5 are strawberries.
,• 1/4 of the strawberries are chocolate covered.
We have to find 1/4 of 2/5, let's multiply these fractions.
\(\frac{1}{4}\cdot\frac{2}{5}=\frac{2\cdot1}{4\cdot5}=\frac{2}{20}=\frac{1}{10}\)Therefore, 1/10 of the strawberries are covered.Move the constant to the right.
3x + 87
− 12 = 0
3x + 87
=
Answers
Answer:
3x+87=12
This is the answer
Answer:
dtcttr57uhiiiiiiiioii
Select the correct answer from each drop-down menu. James needs to clock a minimum of 9 hours per day at work. The data set records his daily work hours, which vary between 9 hours and 12 hours, for a certain number of days. {9, 9.5, 10, 10.5, 10.5, 11, 11, 11.5, 11.5, 11.5, 12, 12}. The median number of hours James worked is . The skew of the distribution is .
Answers
Answer:
Kindly check explanation
Step-by-step explanation:
Given the data:
{9, 9.5, 10, 10.5, 10.5, 11, 11, 11.5, 11.5, 11.5, 12, 12}
The median of the distribution can be found using the formula :
Median = 0.5(n + 1) th term
n = number of terms = 12
Median = 0.5(12 + 1) th term
Median = 0.5(13)th term = 6.5th term
Select the 6th and 7th term = 11, 11
(11 + 11) /2 = 22/2 = 11
Hence, the median = 11
The distribution is negatively skewed, This was inferred from the shape of a distribution having a long shape positioned on the left side of the distribution.
The peak values are the 8th, 9rh and 10th values with only two values below in the right direction with the majority on the left.
Answer:
The median number of hours James worked is 10 . The skew of the distribution is negative .
Step-by-step explanation:
Which function has a constant additive rate of change of -14?
y
-12
21
-1.5
-11
11
-2
-10
14
-2.5
-9
17
Answers
The function which has the constant rate of change as -1/4 is the function(i) whose graph passes through the points (-2,2) and (2,1).
The "Rate-Of -Change" of a graph is defined as a measure of how much one variable changes in relation to another variable. It is calculated by finding slope of line which represents the graph.
The graph of function(i) is a coordinate-plane with a straight line with a negative slope and line passes through (-2, 2) and (2, 1).
The equation is written as :
⇒ y - 1 = [(1-2)/(2+2)]×(x -2),
⇒ y - 1 = (-1/4)×(x -2),
⇒ y = -(x/4) + 1/2 + 1,
⇒ y = -x/4 + 3/2,
Therefore, the constant-additive rate of change = -1/4.
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The given question is incomplete, the complete question is
Which function has a constant additive rate of change of -1/4?
The graph of function(i) is a coordinate plane with a straight line with a negative slope. The line passes through (-2, 2) and (2, 1).
The graph of function(ii) is a coordinate plane with a curved line passing through (-1, 2), (0, -1), the minimum at (2, -2), and (4, -1).
Please help ASAP, thank you
Answers
Answer:
Day 1: 100 bacteria
Day 2: 100 + 90% = 190 bacteria
Day 3: 190 + 90% = 361
Day 4: 361 + 90% = 685.9
Day 5: 685.9 + 90% = 1303.21
Sierra left $17.75 as a tip for a waiter. This was 20% of the bill before the tip. How much was her total bill before the tip
Answers
Answer:
$31.95 Before tip.
Step-by-step explanation:
I believe you should just add 17.75+80%=31.95
Check: 88.75 x 0.2 = 17.75
Three years ago , Ahmed was one third of his sister's age. In a years time ,Ahmed's age doubled will match his sisters age. How old is Ahmed now.
Help the due is tomorrow
Answers
in a year's time, ahmed would be x+4 years old and his sister would be 3x+4. we also know that in a year's time ahmed's age doubled will match his sister's age. hence 2(x+4)=3x+4
2x+8=3x+4
4=x
x is the age of ahmed three years ago. hence, now ahmed is 7 years old.
Could someone help me with my math lab
Answers
All the parts shown in the figure are solved.
What is Combination ?A mathematical technique that is used to determine the possible arrangement of the objects when the sequence of selection does not matter.
Part 1
(a) When two dices are rolled the basic outcomes will be =?
Each dice has 6 outcomes and therefore 6* 6 =36
Therefore total 36 basic outcomes.
(b) Yes both the events are equally likely .
Part 2
(a) There can be 11 types of sum from 2 -12 obtained when two dices are rolled
Sum
2 1+1
3 1+2, 2+1
4 1+3, 2+2, 3+1
5 1+4, 2+3, 3+2, 4+1
6 1+5, 2+4, 3+3, 4+2, 5+1
7 1+6, 2+5, 3+4, 4+3, 5+2, 6+1
8 2+6, 3+5, 4+4, 5+3, 6+2
9 3+6, 4+5, 5+4, 6+3
10 4+6, 5+5, 6+4
11 5+6, 6+5
12 6+6
(b) They are not equally likely as the sum obtained is dependent on both the dice rolling.
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Which ordered pair could replace the missing value and create a function?
Answers
Given that the ordered pair makes the function and these pairs are
[ (2,5), (7,1), (5,9), (8,0), ( , ) ]
Since the function is the one that has the input having only one output.
Hence, the given options (3,4) is correct.
Therefore the ( 3, 4 ) is the pair.
The absolute value function graphed below can also be represented by a piecewise function.
Determine which function is represented by the graph.
please help!!!
Answers
Answer:
Step-by-step explanation:
Trust me
Give the location of Madrid as an ordered pair (x, y).
