When the negative exponent is outside the parentheses indicates that you need to take the inverse of the base number, or put it in the denominator.
Given:
when the negative exponent is outside the parentheses.
suppose (3)^-3 = 1/3^3.
Example:
(7)^-3
Take the inverse of base number and put it in denominator.
= 1/7^3
So the negative is outside the parenthesis is tells that the negative sign must be removed.
Therefore when the negative exponent is outside the parentheses indicates that you need to take the inverse of the base number, or put it in the denominator.
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Can we form a triangle with length 4cm 5cm 9cm?.
Answer:
No. It is not possible to construct a triangle with lengths of its sides 4cm, 5cm and 9cm because the sum of two sides is not greater than the third side:
5 + 4 is not greater than 9.
Step-by-step explanation:
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What is the solution to this system of linear equations 3x 2y 14 5x y 32?.
The solution to the system of linear equation is (x, y) = (6, 2)
What in mathematics is a linear equation?
A linear equation is a first-order (linear) term plus a constant in the algebraic form y=mx+b, where m is the slope and b is the y-intercept.Sometimes, the aforementioned is referred to as a "linear equation of two variables," where x and y are the variables.3x - 2y = 14 (1)
3x - 2y = 14 (1)5x + y = 32 (2)
From (2)
y = 32 - 5x
Substitute y = 32 - 5x into (1)
3x - 2y = 14 (1)
3x - 2(32 - 5x) = 14
3x - 64 + 10x = 14
13x = 14 + 64
13x = 78
x = 78/13
x = 6
Substitute x = 6 into (2)
5x + y = 32 (2)
5(6) + y = 32
30 + y = 32
y = 32 - 30
y = 2
(x, y) = (6, 2)
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The complete question is -
What is the solution to this system of linear equations? 3x – 2y = 14 5x + y = 32 (3, 5) (6, 2) (8, –1) (14, –18)
Answer:
(6,2)
Step-by-step explanation:
i just did the
Translate: the sum of 6 and the quotient of 3 and a number
Answer:
6 + [tex]\frac{3}{y}[/tex]
Step-by-step explanation:
"Sum of" means "this number plus whatever follows." In other words: 6 + ?.
"The quotient of 3 and a number" means 3 divided by said number, so:
6 + 3/x
Can 3 cm 4 cm 5 cm be a triangle?.
Yes, 3 cm 4 cm 5 cm can be a scalene triangle. It is obvious that the lengths of each of the triangle's sides varies from one another.
Define scalene triangle.A scalene triangle is a triangle with three different side lengths and three different angle measurements. The total of all interior angles, however, is always equal to 180 degrees. The roots and trunks of the brachial plexus and the third segment of the subclavian artery exit the neck through the scalene triangle, also referred to as the interscalene triangle, which is situated laterally at the base of the neck. A triangle with three uneven sides is referred to as a scalene triangle.
Given,
Sides 3cm, 4 cm and 5 cm
The sides of a triangle are 3 cm, 4 cm, 4 cm, and 5 cm in length. It is obvious that the lengths of each of the triangle's sides varies from one another. As a result, the triangle in question is a scalene triangle.
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what is the probability that the mean time for the sample of 150 returns for this year is greater than 92?
The probability that the mean time for the sample of 150 returns for this year is greater than 92 is 0.95907
Given,
Mean, [tex]\overline x[/tex] = 90 minutes
standard deviation, [tex]\sigma[/tex] = 14 minutes
to find probability that mean for sample 150 returns is greater than 92, [tex]P(\overline x > 92)[/tex]
This means we need to find probability the related z-value of expected mean
where, [tex]z=\frac{\overline x-\mu_x}{\sigma_x}[/tex]
Here,
[tex]\mu_x[/tex]= expected mean =92
[tex]\sigma_x[/tex]=standard deviation for new sample i.e. 150
[tex]\sigma_x[/tex] can be calculated by formula,
[tex]\sigma_x=\frac{\sigma}{\sqrt{n}}\\\\\sigma_x=\frac{14}{\sqrt{150}}\\\\\sigma_x=1.143[/tex]
Now,
[tex]z=\frac{90-92}{1.143}\\\\z=-1.749[/tex]
Now,
[tex]P(\overline x > 92)\\\\=P(z > -1.749)\\\\=1-P(z < -1.749)\\\\=1-0.04093\\\\=0.95907[/tex]
In above calculation, the z-value is determined from the z-table.
Thus, the probability that mean is greater than 92 is 0.95907
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Your question is incomplete, please complete the question.
What is the common ratio in the geometric sequence 324 108 36?.
The common ratio is 1/3.
What is common ratio?
In a geometric sequence, ratio of each term to its immediately preceding term is always constant and is known as common ratio.
The common ratio is expressed as the value r. It can be calculated by taking any term in the sequence and dividing it by the term before it.
Here, 108/324 = 1/3 ( it is a constant ratio )
Hence it is a geometric sequence and common ratio is 1/3.
Next terms can be found by multiplying the common ratio with the previous term.
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How do you solve an inequality with two signs?.
First, graph the border of the inequality's solution set using a dashed or solid line depending on the inequality.
There is a significant exception to the general rule that inequality manipulation is quite similar to equation manipulation. An inequity still exists if the same amount is added to both sides. The inequality still holds true if the same amount is subtracted from each side. An inequality still holds true if both sides are multiplied or divided by the same positive value. However, if both sides of an inequality are multiplied or divided by a negative value, the inequality is no longer valid.
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Sand falls from an overhead bin and accumulates in a conical pile with a radius that is always three times its height. Suppose the height of the pile increases at a rate of 2 cm/s when the pile is 14 cm high. At what rate is the sand leaving the bin at that instant? Let V and h be the volume and height of the cone, respectively. Write an equation that relates V and h and does not include the radius of the cone. (Type an exact answer, using π as needed.) Differentiate both sides of the equation with respect to t. dV dt (Type an exact answer, using π as needed.) dh dt The sand is leaving the bin at a rate of (Type an exact answer, using π as needed.)
Sand is leaving at the rate of 3528 cm³/s when Sand collects in a conical pile with a radius that is always three times.
Given that,
Sand collects in a conical pile with a radius that is always three times its height as it falls from an overhead container. When the pile is 14 cm high, assume that the height of the pile is growing at a rate of 2 cm/s.
We have to find how quickly is the sand currently leaving the bin. Let V and h represent the cone's volume and height, respectively. Create an equation that connects V and h but leaves out the cone's radius.
We know that,
From given r = 3 × h
Get an equation for the cone's volume in terms of height first.
v = 1/3 × π × r² × h
v = 1/3 × π × (3 × h)² × h
v = 1/3 × π × 9 × h³
Take derivative of both sides
d/dt { v = 3 × π × h }
dv/dt = 9 × pi × h² × dh/dt
In your problem,
dh/dt =2cm/sd
h = 14 cm
Plug in and solve for dh/dt
dv/dt = 9× π × 196 m² × 2
dv/dt = 3528 π cm³/s
Therefore, Sand is leaving at the rate of 3528 cm³/s when Sand collects in a conical pile with a radius that is always three times.
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an unbiased coin is tossed 15 times. in how many ways can the coin land tails either exactly 9 times or exactly 2 times?
Using Combination formula,
Total 5,110 ways are possible that the coin land tails either exactly 9 times or exactly 2 times
Combinations:
If the order of events is not important, the concept of combination determines the number of ways in which a sequence of events can be obtained. I need to find the total number of expected events and the total number of possible events.
We have given that,
A unbiased or fair coin is tossed 15 times.
let us consider two events
A : the coin land tails exactly 9 times
B : the coin land tails exactly 2 times
Now, for calculating the required result we use
Combination formula,
ⁿCₓ= n! /x! (n-x)!
here, n= 15
the number of ways that exactly 9 times coin land as tails (n(A))= ¹⁵C₉ = 15! /9!× 6!
= 15×14×13×12×11×10×9!/ 9!(6×5×4×3×2×1)
= (15×14×13×12×11×10)/( 6×5×4×3×2×1 )
= (15×14×13×11)/6 = 7× 5× 13×11 = 5005
the number of ways that exactly 2 times coin land as tails (n(B))= ¹⁵C₂= 15! /2!× 13!
= 15×14×13!/13!(2)
= 15×14/2 = 15×7 = 105
Now, the number of ways that the coin land tails either exactly 9 times or exactly 2 times
= n(A) + n(B) = 5005 + 105
= 5110
Hence , total required ways are 5,110.
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Highway safety engineers test new road signs hoping that increased reflectivity will make them more visible to drivers. Volunteers drive through a test course with several of the new- and old-style signs and rate which kind shows up the best. a) Is this a one-tailed or a two-t ailed test? Why? b) In this context, what would a Type I error be? c) In this context, what would a Type II error be? d) In this context, what is meant by the power of the test? e) If the hypothesis is tested at the 1 % level of significance instead of 5%, how will this affect the power of the test? f) The engineers hoped to base their decision on the reactions of 50 drivers, but time and budget constraints may force them to cut back to 20. How would this affect the power of the test? Explain.
Volunteers drive through a test course with several of the new and old style signs the rate of the one-tailed test shows up the best.
a) This is a one-tailed test, on the grounds that the researchers are attempting to prove that the higher reflectivity will make the signs more visible to drivers. The visibility will be equal to or less than the null hypothesis, while the visibility will be greater in the alternative hypothesis.
b) A type I error is rejecting the null hypothesis when it ought not to be rejected.
c) The failure to reject the null hypothesis when it should be rejected is a type II error.
d) The power of the test is the probability of dismissing the invalid speculation when it is indeed false.
e) When the significance level is decreased, the test's power decreases. The hypothesis test's acceptance region grows as the level of significance is decreased, increasing the probability that the null hypothesis will not be rejected. A Type II error is more likely to occur because the null hypothesis is less likely to be rejected.
f) The test's power will decrease when the sample size is reduced. The acceptance region will grow larger and the probability of rejecting the null hypothesis will decrease with a smaller sample size.
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Money is invested at two rates of interest. One rate is 8% and the other is 2%. If there is $800 more invested at 8 % than at 2 %. find the amount invested at each
rate if the total annual interest received is $490. Let x = amount invested at 8 % and y = amount invested at 2 %. Then the system that models the problem is
x=y+800
Solve the system by using the method of addition.
0.08x+0.02y = 490
8% is
2% is
Amount invested at rate 8% is, $5060.
Amount invested at rate 2% is, $4260.
What is addition method?
Using the Addition Method to Solve Systems of Equations in Two Variables. The addition approach, often known as the elimination method, is a third strategy for resolving linear equation systems. With this approach, we combine two components that share the same variable but have opposing coefficients, resulting in a sum of zero.
Let, money is invested at two rates of interest.
One rate is 8% and the other is 2%.
There is $800 more invested at 8 % than at 2 %.
Total annual interest received is $490.
Let x = amount invested at 8 % and
y = amount invested at 2 %.
So,
x - y = 800 ..(1)
0.08x + 0.02y = 490 ..(2)
Multiply equation (1) by -8 and equation (2) by 100
-8x + 8y = -6400 ..(3)
8x + 2y = 49000 ..(4)
Adding equation (3) and (4)
(-8x + 8y) + (8x + 2y) = -6400 + 49000
10y = 42600
y = 4260
Plug y = 4260 in equation (1)
x - 4260 = 800
x = 800 + 4260
x = 5060
Hence, amount invested at 8% = $5060.
amount invested at 2% = $4260.
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What 2 facts can you double to find 8 * 4?.
The two facts that might be multiplied by two to get the result of eight and four are (4 and 4)
Given,
The two facts of a doubled number can be written as: 2 b = c.
Here,
The value of c is (84) = 32.
After entering the values into the equation, find b.
2 × b = c
2 × b = 32
Divide through by 2
b = 32 / 2
b = 16
Therefore, (4 4) is a pair of facts that can be doubled.
2(4 × 4) = 2(16) = 32
That is,
The two facts that might be multiplied by two to get the result of eight and four are (4 and 4)
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Sigma-32 is a sigma factor, which mediates response to heat shock. What is the most likely effect in an e. Coli strain which has a mutation in the gene encoding sigma-32?.
According to the sigma factor, the gene encoding sigma-32 states that Genes usually unregulated by heat shock will be less responsive to heat treatment.
Sigma factor:
Sigma factor defines that a detachable polypeptide subunit of RNA polymerase that facilitates the initiation of transcription by recognizing specific DNA promoter sites.
Given,
Sigma-32 is a sigma factor, which mediates response to heat shock. What is the most likely effect in an e. Coli strain.
Here we need to determine in which has a mutation in the gene encoding sigma-32.
While we looking into the given question, Sigma-32 is a sigma factor, which mediates response to heat shock.
Basically, the genes usually unregulated by heat shock will be less responsive to heat treatment.
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identify a set of test cases that satisfies the all defs criterion, for variables x, k and y in method fun. use trace tables to show how the def-use pairs are covered.
Thus, the minimum time taken for a car from start to end of the assembly unit is 14.
Algorithm for assembly line scheduling using dynamic programming:-
1. We need following parameter:
· S[i,j]: The jth station on ith assembly line
· a[i,j]: Time required at station S[i,j], because every station has some dedicated job that needs to done. · e[i]: Entry time of product on assembly line i [here i= 1,2] · x[i]: Exit time from assembly line i· t[i,j]: Time required to transit from station S[i,j] to the other assembly line.
2. Further in dynamic programming we have to ways to do so:
· Reach station S1j from S1,j-1 (same assembly line).
· Reach station S1j from S2,j-1 (switch from second assembly line).
3. Calculate minimum time to leave.
4. After that find the minimum time to leave the previous two stations.
5. Then combining it with the time spent on station.
6. calculate the leaving time of previous stations.
7. Finally, we need two tables to store the partial results that are calculated for each station in an assembly line.
Example:- a = [[5,4,3], [2,3,7]], t = [[0,2,2], [0,1,1]]
e1 = 3, e2 = 2
x1 = 3, x2 = 4
T1[0] = e1 + a[0][0] = 3+5 = 8
T2[0] = e2 + a[0][1] = 2+2 = 4
T1 = [8, 0, 0]
T2 = [4, 0, 0]
i = 1
T1[1] = min(T1[1-1] + a[0][1], T2[1-1] + t[1][1] + a[0][1])
= min(8+4, 4+1+4) = min(12,9)
T1[1] = 9
T2[1] = min(T2[1-1] + a[1][1], T1[1-1] + t[0][1] + a[1][1])
= min(4+3, 8+2+3) = min(7,13)
T2[1] = 7
T1 = [8, 9, 0] T
2 = [4, 7, 0]
i = 2
T1[2] = min(T1[2-1] + a[0][2], T2[2-1] + t[1][2] + a[0][2])
= min(9+3, 7+1+3) = min(12,11)
T1[2] = 11
T2[2] = min(T2[2-1] + a[1][2], T1[2-1] + t[0][2] + a[1][2])
= min(7+7, 9+2+4) = min(14,15)
T2[2] = 14​
T1 = [8, 9, 11]
T2 = [4, 7, 14]
ans = min(T1[n-1]+x1, T2[n-1]+x2)
= min(11+3, 14+4) = min(14, 18)
ans = 14
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how to graph (1, 1 1/3) and (2, 2/3) on a cartesian plane? kindly explain how
The points have two coordinates, (x, y)
The first coordinate, x represents the distance from zero on the x-axis,The second coordinate, y represents the distance from zero on the y-axis.To plot a point, mark x coordinate on the x-axis, y-coordinate on the y-axis, then add imaginary lines parallel to x- and y-axis, the intersection of those lines is the point to be plotted.
The points we have:
(1, 1 1/3) and (2, 2/3)See attached, it should be clear visually.
JUSTIFY REASONING You are evaluating a conditional statement in which the hypothesis is true, but the conclusion is false. Is the inverse of the
statement true or false? Justify your argument.
Instructions
Answer: The statement is false.
Step-by-step explanation: This is because inverse means that you just negate both so the hypo is now false and the conclusion is now true. Which makes this statement false.
Answer: true
Step-by-step explanation:
The temperature of a pot of water was 180°F and it changes at a rate of -2.25°F per minute.
A. What is the temperature after 20 minutes?
B. How many minutes will it take the water to cool from 180°F to 100°F?
(Please help me! I'm kinda stuck on this! )
The temperature equation is T(x) = 180 - 2.25*x, using that we will see:
A) after 20 minutes the temperature is 135
B) the temperature will be 100 after 35.56 minutes.
What is the temperature after 20 minutes?We know that the initial temperature is 180°F and it changes at a rate of -2.25°F per minute, so after x minutes, the temperature is:
T(x) = 180 - 2.25*x
A) after 20 minutes, the temperature is:
T(20) = 180 - 2.25*20 = 135
The temperature after 20 minutes is 135
B) How long will it take to go from 180°F to 100°F?
We need to solve the equation:
T(x) = 100
180 - 2.25*x = 100
180 - 100 = 2.25*x
80/2.25 = x
35.56 = x
So after 35.56 minutes, the temperature will be 100°F
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How do you find the slope-intercept form of a perpendicular line?.
The equation for the slope intercept form of a perpendicular line will be;
⇒ y - y₁ = - 1/m₂ (x - x₁)
Where, m₂ is slope of perpendicular line.
What is Equation of line?
The equation of line in point-slope form passing through the points (x₁ , y₁) and (x₂, y₂) with slope m is defined as;
⇒ y - y₁ = m (x - x₁)
Where, m = (y₂ - y₁) / (x₂ - x₁)
Given that;
The slope-intercept form of a perpendicular line.
Now,
We know that;
The product of slopes of perpendicular lines are - 1.
That is, Let m₁ be the perpendicular line and m₂ is there corresponding line.
Then, m₁ m₂ = - 1
⇒ m₁ = - 1/m₂
Thus, The equation for the slope intercept form of a perpendicular line will be;
⇒ y - y₁ = - 1/m₂ (x - x₁)
Where, m₂ is slope of perpendicular line.
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To simulate a toss of a coin we let the digits 0, 1, 2, 3, and 4 correspond to a head and the digits 5, 6, 7, 8, and 9 correspond to a tail. Consider the following game: We are going to toss the coin until we either get a head or we get two tails in a row, whichever comes first. If it takes us one toss to get the head we win $2, if it takes us two tosses we win $1, and if we get two tails in a row we win nothing. Use the following sequence of random digits: 12975 13258 45144The estimated number of tosses in a single trial of the game is?A)2.0B)15/9C)15/11D)11/7E)7/11
As per the given probability, the estimated number of tosses in a single trial of the game is 7/11
Probability:
Basically, the term Probability refers the possibility of happening the particular event.
Given,
Here we have to simulate a toss of a coin we let the digits 0, 1, 2, 3, and 4 correspond to a head and the digits 5, 6, 7, 8, and 9 correspond to a tail. And consider the following game: We are going to toss the coin until we either get a head or we get two tails in a row, whichever comes first. If it takes us one toss to get the head we win $2, if it takes us two tosses we win $1, and if we get two tails in a row we win nothing.
While we looking into the given question we have identified that, the total number possible events is 11.
And we have to find the single trial of the game that is 7.
So, the estimated number of tosses in a single trial of the game is written as,
=> 7/11
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In 1975, a Camaro cost $4,000. In 2022, a Camaro cost $25,000. What was the percent of change from
1975 to 2022 in the cost of a Camaro? Show work. Label your answer as increase or decrease.
Please help me
I give 20 points
Thanks
Answer:
420%. and it will be labeled as increase.
Answer:
525% increase
Step-by-step explanation:
The percentage increase formula is:
(new number - old number/old number) × 100
We can now put in our numbers to substitute the variables.
(25000-4000/4000) × 100
=(21000/4000) × 100
=(5.25) × 100
= 525
Therefore, the answer is a 525% increase.
Hope this helps and good luck on your homework!
3. Find the measure of the angle indicated.
Answer:
94°
Step-by-step explanation:
Interior angles:
m<C = 49°
m<D = ?
Angles CBD and CBY are supplementary.
m<CBD = 180° - 143° = 37°
m<C + m<D + m<CBD = 180°
49° + m<D + 37° = 180°
m<D = 94°
What is f/x )= x 2 on a graph?.
The graph of given function and its explanation is explain below.
What is parabola?Parabola a bend shaped by the convergence of a cone with a plane lined up with a straight line in its surface : a bend framed by a moving so that its separation from a proper point is equivalent to its separation from a decent line. : something that is bowl-formed. allegorical.
According to question:Graph of the function f(x)=x^2 which is a parabola is
One of the ways of charting this is to involve plug in a couple of x-esteems and find out about the shape. Since the x qualities continue to get squared, there is a remarkable increment on one or the other side of the y-pivot. You can see this by connecting a couple of values:
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a restaurant offers salads with 2 types of lettuce, 3 different toppings, and 3 different dressings. how many different salads could be ordered?
Answer:
36 different salads can be ordered
Step-by-step explanation:
M is the centroid of the triangle.
CM=7, PM=10, & BQ=18
BM=[ ?]
Although the p-value does not give us the probability that the null hypothesis is true, it does suggest the likelihood that it is true. Match the p-value to the strength of the case in favor or the null hypothesis.
The p-value to the strength of the case in favor of the null hypothesis is
0.20 → A good chance H0 is correct.
0.05 → A slight chance H0 is true.
0.01 → Very poor chance H0 is true.
0.001→ Extremely poor chance H0 is true.
The p-value lets you know how likely the information you have noticed is to have happened under the invalid speculation if the p-value is underneath your edge of importance (commonly p < 0.05), then, at that point, you can dismiss the null hypothesis, yet this doesn't guarantee to imply that your elective speculation is valid.
P-value is frequently used to advance believability for studies or reports by government offices. For instance, the United States Census Bureau specifies any examination with a p-value more noteworthy than 0.10 should be joined by an explanation that the thing that matters isn't measurably quite the same as zero The Census Bureau likewise has guidelines set up specifying what p-values are acceptable for different distributions
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Using a long rod that has length , you are going to lay out a square plot in which the length of each side is μ . Thus the area of the plot will be μ2. However, you do not know the value of μ, so you decide to make n independent measurements X1, X2, . . . , Xn of the length. Assume that each Xi has mean (unbiased measurements) μ and variance σ2.
Proved that X² is not an unbiased estimator for μ²
Given,
The length of the square plot = μ
The area of the square plot = μ²
You have to make the independent measurements X₁, X₂,.....Xₙ
Variance = σ²
a) We have to show that X² is not an unbiased estimator for μ²
Here,
E(X²) = V(X) + [E(X)]²
= σ²/n + μ² ≠ μ²
Then,
X² is not an unbiased estimator for μ²
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if a(t) is the amount of the investment at time t for the case of continuous compounding, write a differential equation satisfied by a(t). Let the rate be r. Then the amount of the investment must satisfy the differential equation A′ (t) = r A (t)
The initial condition satisfied by A (t) is its value at t = 0. At t=0, no compounding has occurred yet, and the amount is equal to the original amount invested, called the principal. If the principal is P, then the initial condition is A (0) = P
The differential equation that satisfies the given amount of investment is A'(t) = rA(t)
Let the rate of investment = r
The amount of investment must satisfy the differential equation
A'(t) = r A(t)
Initial condition is the value of A(t) at t = 0
No compounding will be applied at t = 0 and the amount is equal to the original amount invested, known as principal.
If the Principal amount is P, then the initial condition is A(0) = P
Hence, the differential equation is A'(t) = rA(t)
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Make a sketch of a linear relationship with slope of 3 that is not a proportional relationship.
Sketch is attached in the graphical form to illustrate the linear relationship with slope of 3 that is not the proportional relationship.
What is linear relationship ?A statistical word is used to express the straight-line relationship between two variables which is a linear relationship (or linear association). A straight line connecting the variable and the constant can be used to represent a linear relationship graphically, or the dependent variable can be determined by multiplying the independent variable by the slope coefficient and a constant.
A linear non - proportional relationship could be written with as a slope - intercept with a non - zero intercept value. One such relationship with a slope of 3 can be expressed thus :
y = 3x + 5
Here, the linear relationship, has a slope value of 3 and an intercept value of 5. The function cannot be proportional due to the presence of a non-zero constant
Hence, Using a graph plotter, the sketch of the given linear relationship is attached.
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1212 members of a wedding party are lining up in a row for a photograph. (1) how many ways are there to line up the 1212 people?
The total number of ways to line up 12 people are 479,001,600.
We have 12 members of wedding party. We need to make arrangements for 12 people to line up for a photograph.
If we have total n objects and we need to arrange r objects then it can be done in [tex]^nP_r[/tex] ways which is equivalent to =(n!/(n-r)!)
Now, here we have 12 members for photograph, we are free to choose any of members from 12 members.
So, this can be done in [tex]^1^2P_1_2[/tex] ways which is equivalent to =(12!/(12-12)!)=12!/0!
We know that 0!=1
Therefore,12!/0!=12!=479,001,600
Hence, total number of ways are 479,001,600.
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consider two random variable x1, x2, which are uniformly distributed in the triangle. now suppose that y1 y2
The difference between 2X₁ and X₁+X₂ is N(0,6σ²)
According to the Central Limit Theorem, any large sum of the independent, identically distributed(iid) random variables is approximately Normal.
We know very well that the Normal distribution is defined by two parameters, the mean is μ , and the variance is σ² and written as X=N(μ,σ²).
Using the properties of normal random variables, we get
if X=N(μ₁,σ₁²)and Y=N(μ₁,σ₁²) are two independent identically distributed random variables then
the sum of normal random variables is represented by
=>X+Y=N(μ₁+u₂,σ₁²+σ₂²)
and the difference of normal random variables is represented by
=>X-Y =N( μ₁-u₂,σ₁²+σ₂²)
When Z=aX+bY, the linear combination of X and Y is given by
=>Z=N(aμ₁+bu₂,a²σ₁²+b²σ₂²)
Similarly, When Z=aX , the product of X is given by
=>Z= N(aμ₁,a²σ₁²)
We need to find 2X₁
Thus, following the property of multiplication, we get
=>2X₁=N(2μ,2²σ²)
=>2X₁=N(2μ,4σ²)
and following the property of addition,
X₁+X₂=N(μ+μ,σ²+σ²)
And the difference between the two is given by
2X₁-(X₁+X₂)=N(2μ-2μ,2σ₁²+4σ₂²)
=>2X₁-(X₁+X₂)=N(0,6σ²)
Hence, the required difference is N(0,6σ²).
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(Complete question) is:
consider two random variable X₁, X₂, which are uniformly distributed in the triangle. now suppose that Y₁ ,Y₂ are two random variables and If X₁ = N(μ, σ₂) and X₂ =N(μ, σ₂) are iid normal random variables, then what is the difference between 2 X₁ and X₁ + X₂?
