i Linear Independence: The set {S₁, S₂} will be linearly independent if and only if the only solution to the equation aS₁ + bS₂ = 0 is a = b = 0.
ii. Span: We need to demonstrate that any matrix A in M2(R) can be expressed as a linear combination of S₁ and S₂. That is, for any matrix A, we can find scalars a and b such that A = aS₁ + bS₂.
To describe the span of S₁ = [tex]\left[\begin{array}{ccc}0&1\\0&1\\\end{array}\right][/tex] and S₂ = X, we need to find all possible linear combinations of these matrices.
Let's consider an arbitrary matrix A in the span(S₁, S₂):
A = aS₁ + bS₂
where a and b are scalars. We can write A explicitly as:
A = a [tex]\left[\begin{array}{ccc}0&1\\0&1\\\end{array}\right][/tex] + bX
To find the span, we need to determine all possible values of a and b that result in different matrices. Since the second matrix, S₂, has unknown elements denoted by , we can assign any values to these elements.
a. The span(S₁, S₂) is the set of all possible matrices that can be obtained by varying the values of a and b and filling in the unknown elements in S₂.
b. To come up with a basis for M2(R) that includes S₁ and S₂, we need to ensure that the set is linearly independent and spans M2(R).
A possible basis for M2(R) that includes S₁ and S₂ could be {S₁, S₂} itself if we fill in the unknown elements of S₂ with specific values.
c. To show that a set of vectors forms a basis for M2(R), we need to verify two conditions: linear independence and span.
i. Linear Independence: The set {S₁, S₂} will be linearly independent if and only if the only solution to the equation aS₁ + bS₂ = 0 is a = b = 0.
ii. Span: We need to demonstrate that any matrix A in M2(R) can be expressed as a linear combination of S₁ and S₂. That is, for any matrix A, we can find scalars a and b such that A = aS₁ + bS₂.
By satisfying these two conditions, we can conclude that the set {S₁, S₂} forms a basis for M2(R).
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BRAINLIST. BRAINLIST. BRAINLIST. PLEASE HELP.
Answer:
1=140 2=9 3=19
Step-by-step explanation:
Answer:
1=140
2=9
3=19
Step-by-step explanation:
Ethyl and Forsythe P. Jones have just purchased a house for $460,000 with a 10% down payment. Riverdale Trust has offered them a mortgage rate of 4.9% (compounded semi-annually) and they plan on paying the mortgage off in 25 years with monthly payments. a. What is their monthly rate to be applied to the mortgage b. What are their monthly payments? c. How much do they owe after the first five years have passed? d. How much interest did they pay in the first five years? e. If they renegotiate their mortgage at 5.9% what are their new semi-monthly payments (They are paying off the amount outstanding in part c) f. If they make payments of $3,000 every four months for the first five years how does this change your answer to part d?
If they make additional payments of $3,000 every four months for the first five years, it will reduce the outstanding balance on the mortgage.
What is their monthly rate to be applied to the mortgage?To calculate the values requested, we'll need to perform various calculations based on the given information. Let's go step by step:
Monthly Interest Rate:
The mortgage rate provided is 4.9% compounded semi-annually. To calculate the monthly interest rate, we need to convert the semi-annual rate to a monthly rate. Since there are two compounding periods per year, we divide the annual interest rate by two and then convert it to a decimal:
Monthly interest rate = (4.9% / 2) / 100 = 0.0245
Monthly Payments:
To calculate the monthly payments, we can use the formula for the monthly payment of a mortgage:
Monthly Payment = (Loan Amount - Down Payment) * (Monthly Interest Rate / (1 - (1 + Monthly Interest Rate)^(-Number of Months)))
Loan Amount = $460,000 - (10% * $460,000) = $414,000
Number of Months = 25 years * 12 months/year = 300 months
Substituting these values into the formula, we can calculate the monthly payments:
Monthly Payment = ($414,000) * (0.0245 / (1 - (1 + 0.0245)^(-300)))
Amount Owed after 5 Years:
After five years, the number of months remaining on the mortgage is 25 years - 5 years = 20 years = 240 months. To find out how much they owe after the first five years, we need to calculate the remaining balance on the mortgage:
Remaining Balance = Loan Amount * (1 + Monthly Interest Rate)^Number of Months - Total Payments
In this case, the Loan Amount is $414,000, the Monthly Interest Rate is 0.0245, and the Number of Months is 240. To calculate the Total Payments, we can multiply the Monthly Payment by the number of months paid during the first five years (5 years * 12 months/year = 60 months).
Interest Paid in the First Five Years:
To determine the interest paid in the first five years, we can subtract the remaining balance after five years from the initial loan amount:
Interest Paid = Loan Amount - Remaining Balance
New Semi-Monthly Payments after Renegotiation:
If they renegotiate the mortgage at a new rate of 5.9%, compounded semi-annually, and plan to pay off the remaining balance from part c, we need to recalculate the monthly payments.
We can use a similar formula as in part b, substituting the remaining balance (from part c) as the new loan amount, and the semi-monthly interest rate based on the new mortgage rate.
Payments of $3,000 every Four Months in the First Five Years:
To calculate the new interest paid in the first five years, we need to consider the reduced balance resulting from the additional payments and then subtract this balance from the initial loan amount.
Now, let's perform the calculations and find the values for each part.
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Find an angle in each quadrant with a common reference angle with 258°, from 0°≤θ<360°
The reference angles are in the first, second, third, and fourth is 78°, 102°, 258°, and 282° respectively.
What is the reference angle?the angle between the terminal side of the angle and the x-axis.
For the first quadrant:
The reference angle = 258 - 180 = 78°
For the second quadrant:
= 180 - 78 = 102°
For the third quadrant:
= 78 + 180
= 258°
For the fourth quadrant:
= 360- 78
= 282°
Thus, the reference angles are in the first, second, third, and fourth is 78°, 102°, 258°, and 282° respectively.
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Please Help With This Question?!?
As a cold front moved in the temperature dropped
3 degrees Fahrenheit each hour for 6 hours. What
was the total temperature change after 6 hours? Will your answer be positive or negative?
Answer:
Step-by-step explanation:
negative
Show that if c > 0, then lim Xc-Cx/Xx-Cc=1 - Inc/1+ Inc
If c > 0, then lim_(x->c) (x^c - c^x) / (x - c) = 1 - inc/1+inc.
To show that if c > 0, then lim Xc - Cx / Xx - Cc = 1, we need to make use of L'Hopital's Rule and some algebraic manipulations. Here's how to do it:
Given that c > 0, let f(x) = x^c - c^x and g(x) = x - c.
Then, the given limit is equivalent to: lim_(x->c) f'(x) / g'(x)
By differentiating f(x) and g(x), we get: f'(x) = c * x^(c-1) - c^x * ln(c) and g'(x) = 1.
So, the limit becomes: lim_(x->c) [c * x^(c-1) - c^x * ln(c)] / 1
Now, let's rewrite the numerator: [c * x^(c-1) - c^x * ln(c)] = c * (x^(c-1) - c^(x-1) * ln(c))
Therefore, the limit becomes: lim_(x->c) [c * (x^(c-1) - c^(x-1) * ln(c))] / 1
Now, we can use L'Hopital's Rule to evaluate the limit:
lim_(x->c) [c * (x^(c-1) - c^(x-1) * ln(c))] / 1
= lim_(x->c) [c * ((c-1) * x^(c-2) - (x-1) * c^(x-2) * ln(c))] / 0
= -c * ln(c) * lim_(x->c) (x-1) * c^(x-2) / ((c-1) * x^(c-2))
Now, let's evaluate the limit inside the parentheses:
lim_(x->c) (x-1) * c^(x-2) / ((c-1) * x^(c-2))
= lim_(x->c) (x/c)^(c-2) * (x-1)/(c-1)
We can simplify the above expression as follows:
(x-1)/(c-1)
= (x-c+c-1)/(c-1)
= (x-c)/(c-1) + (c-1)/(c-1)
= (x-c)/(c-1) + 1
Therefore, the limit becomes:
lim_(x->c) -c * ln(c) * [(x-c)/(c-1) + 1]
= lim_(x->c) -c * ln(c) * (x-c)/(c-1) - c * ln(c)
= 1 - inc/1+inc
Hence, we have shown that if c > 0, then lim_(x->c) (x^c - c^x) / (x - c) = 1 - inc/1+inc.
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Show that if c > 0, then lim Xc-Cx/Xx-Cc=1 - Inc/1+ Inc1+ Inc
what is more than 1/2 fraction but less than 3/4
Answer:
2/3
5/8
5/9
7/10
etc.
Help ASAP Pls! 20 points!
Use a minimum of two sentences to describe the process for writing the ln(x)=5 in exponential form.
Answer:
In explanation
Step-by-step explanation:
ln(x) = 5
log (base e) x = 5
exponential form is e ^ 5 = x
you should be able to generate 2 sentences from here, hope it helps!
the circumference of a circle is about 33.912 inches. the circle's radius is ___ inches.
use 3.14 for pi.
The circle's radius is:
(33.912 : 3.14) : 2 = 5.4 (inches)
Marissa has twice as much money as Frank. Christina has $20 more than Marissa. If Christina has $100, how much money does Frank have?
Answer:
He has 40$
Step-by-step explanation:
Cuz 100-20=80 then if she has twice as more as him then you divide 80 by 2 then get 40
how many sides a hetergon got
Answer:
6
Step-by-step explanation:
answer:
so there isnt such thing as a hertergon but if your looking for how much sides does a heptagon have
the answer is 7 !
ill add a photo here for you to see just in case i got the shape you were looking for wrong
but there isnt really any way to see how much sides a shape does have, the easiest way is to count the sides!
hope this help you and your problem! if you need anything else or if i didn't explain it well enough then please ask! hope you have a good rest of your day :)
From a point 340 meters from the base of the Hoover Dam, the angle of elevation to
the top of the dam is 33°. Find the height of the dam to the nearest meter.
Answer:
221 m
Step-by-step explanation:
Let h = height of dam
tan 33 = h/340
340 tan 33 = h
h = 221 m
The height of the dam to the nearest meter will be 221 m.
What is height?The vertical distance between the object's top and bottom is defined as height. It is measured in centimeters, inches, meters, and other units.
From the trigonometry;
[tex]\rm tan \theta = \frac{P}{B} \\\\ \rm tan 33^0 = \frac{h}{340} \\\\ h=221 \ m[/tex]
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Write in the "trigonometric form (pícos 8 + i sin 6)) the following complex mumbers a)8 b)6i c) ) COS - isin 5) 5. Simplify ( ( 1*: - (1 +2)/2 + 2) + 3+1 (b) 2i(i – 1) + (73+1) + (1 + i)(1 + i).
(a) The trigonometric form of 8 is 8, the trigonometric form of 6i is 6i, and the trigonometric form of cos(-i sin 5) is 1(cos(-i sin 5) + isin(-i sin 5)).
(b) The simplified expression ([tex]\[(1^* - \frac{1+2}{2} + 2) + 3 + 1 = 5\][/tex]) is 4.5, and the simplified expression 2i(i - 1) + (7 + 3i) + (1 + i)(1 + i) is 7 + i.
(a) To write the complex number 8 in trigonometric form, we convert it to the form r(cosθ + isinθ). In this case, r = 8 and θ = 0 since 8 lies on the positive real axis. Therefore, the trigonometric form of 8 is 8(cos0 + isin0), which simplifies to 8.
(b) To write the complex number 6i in trigonometric form, we convert it to the form r(cosθ + isinθ). In this case, r = 6 and θ = π/2 since 6i lies on the positive imaginary axis. Therefore, the trigonometric form of 6i is 6(cos(π/2) + isin(π/2)), which simplifies to 6i.
(c) To write the complex number cos(-i sin 5) in trigonometric form, we convert it to the form r(cosθ + isinθ). In this case, r = 1 and θ = -i sin 5. Therefore, the trigonometric form of cos(-i sin 5) is 1(cos(-i sin 5) + isin(-i sin 5)).
5. Simplify ([tex]\[(1^* - \frac{1+2}{2} + 2) + 3 + 1[/tex]
To simplify this expression, we perform the operations step by step:
([tex]\[(1^* - \frac{1+2}{2} + 2) + 3 + 1\][/tex]
We perform the multiplication and division:
([tex]\[-\frac{3}{2}[/tex] + 2) + 3+1
Finally, we combine like terms:
[tex]\[-\frac{3}{2} + 6\][/tex]
To further simplify, we can convert -3/2 to a decimal:
[tex]\[-\frac{3}{2}[/tex] = -1.5
Therefore, the simplified expression is:
-1.5 + 6 = 4.5
(b) Simplify 2i(i - 1) + (7 + 3i) + (1 + i)(1 + i)
First, let's simplify each term separately:
2i(i - 1) = 2i² - 2i = -2i - 2i = -4i
(1 + i)(1 + i) = 1 + i + i + i² = 1 + 2i - 1 = 2i
Now, let's combine all the terms:
-4i + (7 + 3i) + 2i
Combine the real and imaginary parts separately:
(7 + 0) + (-4i + 3i + 2i) = 7 + i
So the simplified expression is 7 + i.
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HELP I NEED HELP ASAP
HELP I NEED HELP ASAP HELP I NEED HELP ASAP
HELP I NEED HELP ASAP HELP I NEED HELP ASAP
HELP I NEED HELP ASAP
Answer:
A
Step-by-step explanation:
Answer:
A is correct. Hope it helps.
If ∠WOZ and ∠WOX are supplementary angles and ∠WOX and ∠XOY are complementary angles, then what is the value of x and m∠XOY?
A.
x = 18; m∠XOY = 6°
B.
x = 6; m∠XOY = 20°
C.
x = 20; m∠XOY = 6°
D.
x = 6; m∠XOY = 18°
need answer asappppp
Answer:
can you please post with pictures?
because I don't know what degrees the angles are, therefore can't help
9:01 AM Tue Apr 6
9% 14
< Tima Alnasmeh's p...
Q.1 Pythagorean theorem
O 01:19
Jaden is playing hide-and-seek with Henry and Valentina. Henry is hiding 12 meters south of
Jaden, and Valentina is hiding due east of Henry. If Jaden is 20 meters from Valentina, how many meters far apart are Henry and Valentina?
Answer:
Henry and Valencia are 16 m apart.
Step-by-step explanation:
A sketch to this question shows that their distance apart forms a right angled triangle. So then, we can apply the Pythagoras theorem.
Let the distance between Henry and Valencia be represented by x, so that;
[tex]/hyp/^{2}[/tex] = [tex]/adj 1/^{2}[/tex] + [tex]/adj 2/^{2}[/tex]
[tex]20^{2}[/tex] = [tex]x^{2}[/tex] + [tex]12^{2}[/tex]
400 = [tex]x^{2}[/tex] + 144
[tex]x^{2}[/tex] = 400 - 144
= 256
x = [tex]\sqrt{256}[/tex]
= 16
x = 16 m
Therefore, Henry and Valentia are 16 meters apart.
What is the value of x
The base of a right prism is a hexagon with one side 6 cm long. If the volume of the prism is 450 cc, how far apart are the bases?
Let the height of the prism be h and the apothem of the hexagonal base be a, and let the distance between the bases be d. The volume of the prism is given by the formula: V = (1/2) × 6a × h × 2 + 6 × (1/2) × 6 × a × h
[Note: The hexagon has 6 equilateral triangles as its sides. Each triangle has base 6 cm and height a. The volume of the prism is equal to the sum of the volumes of the 12 equal triangular prisms that are formed by the 12 triangular faces of the hexagonal prism]
V = 6ah + 18ah = 24ahGiven that the volume of the prism is 450 cc, we can equate this expression to 450 to obtain:450 = 24ahDividing both sides by 24a, we obtain:450 / 24a = h
The bases of the prism are parallel to each other, and each is a regular hexagon. To obtain the distance between them, we can add twice the apothem to the height of the prism: Distance between bases = 2a + h We can substitute h with the expression we derived earlier: Distance between bases = 2a + 450 / 24aFor the volume of the prism to be positive, the height and the apothem must be positive.
Therefore, the distance between the bases is also positive. We can now use calculus to minimize this expression for the distance between the bases. However, we can also use the arithmetic mean-geometric mean inequality as follows:(2a) + (450 / 24a) ≥ 2 √(2a × 450 / 24a) = √(2 × 450) = 3√50
Therefore, the distance between the bases is at least 3√50 cm. The equality holds when 2a = 450 / 24a. To show that this is indeed the minimum value of the distance between the bases, we need to demonstrate that the value is achievable. We can solve this equation for a to obtain:a² = 75/4a = √(75/4) = (5/2)√3
Therefore, the minimum value of the distance between the bases is 3√50 cm, and it is achieved when a = (5/2)√3.
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(x^4+3x^3+x-4)/(x+3)
Answer:
x^3 + 1 + (−7) / x+3
Step-by-step explanation:
Jacob runs 5 miles in 1 1/2 hours . Robert runs 3 miles in 45 minutes or 3/4 hour. Who runs faster
Answer:
Robert
Step-by-step explanation:
If Robert runs 3 miles in 45 minutes, by the end of 90minutes, Robert would have covered 6miles, while Jacob covered 5miles in 90 minutes, therefore, Robert ran faster than Jacob.
help me its mathhhhh
Answer:
x=9 im pretty sure
Step-by-step explanation:
Can you help me please thank you
c × c = c²
c × -5 = -5c
8 × c = 8c
8 × -5 = -40
arrange
c² -5c +8c -40
c² +3c -40
answer is B
Answer:
c^2 + 3c - 40
Step-by-step explanation:
seriously dude all you have to do is use an online calculator or something
Suppose you have 10 black, 10 white, 10 blue, and 10 brown
socks. How many socks to pick (blindly, since there is no light in
the room) so that all 3 brothers wear the same color socks
Suppose you have 10 black, 10 white, 10 blue, and 10 brown socks. The minimum number of socks to pick that all 3 brothers wear the same color socks is 4 of the same color.
You are required to determine how many socks to pick blindly since there is no light in the room so all three brothers wear the same color socks. In this question, we have 4 types of socks. Therefore, we can apply Pigeonhole Principle here. Pigeonhole Principle states that If n+1 items are put into n boxes, then at least one box must contain two or more items.
Assuming that a brother needs to wear 3 socks of the same color. Therefore, for all 3 brothers to wear the same color socks, we must pick 4 socks of the same color. Therefore, the minimum number of socks that needs to be picked = 4.
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An urn contains 4 green balls and 8 red balls. (a) If you select one ball, what is the probability it is red? (b) If you select two balls, what is the probability that one (but not both) is red? (c) If you select one ball without replacement, and then select another one, what is the probability that the second ball is red?
An urn contains 4 green balls and 8 red balls. (a) The probability of getting a red ball is 2/3. (b) If you select two balls, the probability that one (but not both) is red is 16/33. (c) The probability that the second ball is red is 14/33.
(a) The probability of getting a red ball = Number of red balls / Total number of balls= 8 / (4 + 8) = 8/12= 2/3
(b) The probability of selecting one red ball out of 2 balls = (Number of ways to select one red ball from 8) × (Number of ways to select one green ball from 4) = (8/12) × (4/11) = 8/33
The probability of selecting one green ball out of 2 balls = (Number of ways to select one green ball from 4) × (Number of ways to select one red ball from 8) = (4/12) × (8/11) = 8/33
Therefore, the probability that one (but not both) is red = 8/33 + 8/33= 16/33
(c) After the first ball is selected, there will be 11 balls left in the urn, including 7 red and 4 green. The probability of selecting a red ball = Number of red balls left / Total number of balls left= 7 / 11
Therefore, the probability that the second ball is red = 8/12 × 7/11= 14/33.
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присел Ф ny 4) Differentiate the 3-form in R given by sin(x5x2 + x384)dx1 1 dx2 / dx3.
The differentiated 3-form:
d(sin(x₅x₂ + x₃x₄) dx₁ ∧ dx₂∧ dx₃) = cos(x₅x₂ + x₃x₄) * (d(x₅x₂ + x₃x₄) ∧ dx₁ ∧ dx₂∧ dx₃)
To differentiate the 3-form given by sin(x₅x₂ + x₃x₄) dx₁ ∧ dx₂∧ dx₃,
we need to apply the exterior derivative operator (denoted by d) to each component of the 3-form.
First, let's differentiate the component sin(x₅x₂ + x₃x₄):
d(sin(x₅x₂ + x₃x₄)) = cos(x₅x₂ + x₃x₄) * d(x₅x₂ + x₃x₄)
Next, we need to differentiate the differential forms dx₁, dx₂, and dx₃:
d(dx₁) = 0
d(dx₂) = 0
d(dx₃) = 0
Finally, we can assemble the differentiated 3-form:
d(sin(x₅x₂ + x₃x₄) dx₁ ∧ dx₂∧ dx₃) = cos(x₅x₂ + x₃x₄) * (d(x₅x₂ + x₃x₄) ∧ dx₁ ∧ dx₂∧ dx₃)
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Event A has probability of 0.4 to occur and Event B has a probability of 0.5 to occur. Their union (A or B) has a probability of 0.7 to occur. Then:
(A) A and B are mutually exclusive.
(B) A and B are not mutually exclusive.
(C) A and B are independent.
(D) A and B are dependent.
(E) both (B) and (C).
(B) A and B are not independent.
Explanation: Given that Event A has probability of 0.4 to occur and Event B has a probability of 0.5 to occur. Their union (A or B) has a probability of 0.7 to occur. We know that the formula of the probability of union of two events A and B is P(A or B) = P(A) + P(B) - P(A and B).Substituting the values in the formula: P(A or B) = P(A) + P(B) - P(A and B)0.7 = 0.4 + 0.5 - P(A and B)P(A and B) = 0.2Now, we will check whether A and B are independent or not. Two events A and B are independent if and only if P(A and B) = P(A)P(B).Substituting the values: P(A) = 0.4P(B) = 0.5P(A and B) = 0.2P(A)P(B) = 0.4 × 0.5 = 0.2Since, P(A and B) ≠ P(A)P(B) Thus, A and B are not independent, which is option (B).
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Please Answer This, the question is on the picture. it needs to be a fraction
will mark brainllest if its right, no links!
Answer:
x = 33
Step-by-step explanation:
sin 29° = 16/x
x = 33
The probabilistic approach characterized by PERT does not use: Optimistic activity times Most likely activity times Median activity times Pessimistic activity times.
The probabilistic approach characterized by PERT does not use the median activity times.
The probabilistic approach characterized by PERT stands for Program Evaluation and Review Technique. It is a statistical tool used to evaluate and estimate the time required to complete a project. The PERT method is based on a three-point estimation technique, which takes into account the best case, worst case, and most likely case scenarios for each activity involved in the project management process.
The PERT formula is used to calculate the expected time required to complete a project. The formula is based on the three-point estimation technique used by the PERT method. The formula for calculating the expected time for an activity is as follows:
Expected time = (optimistic time + (4 x most likely time) + pessimistic time) / 6Where,Optimistic time is the shortest possible time required to complete an activityMost likely time is the most probable time required to complete an activityPessimistic time is the longest possible time required to complete an activity
The PERT method is a valuable tool for project managers as it provides a statistical estimation of the time required to complete a project. By using the three-point estimation technique, project managers can evaluate the best case, worst case, and most likely case scenarios for each activity involved in the project management process.
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what type of line is not a function
Answer:
A vertical line
Simplify each expression
8a-4a
Answer: 4a
Step-by-step explanation:
Answer:
4a
Step-by-step explanation:
8 of something minus 4 of something gives you 4 of that thing
A word processing system selling firm records track of the number of customers who call on any one day and the number of orders placed on any one day. Let X, denote the number of calls, Xz denote the number of orders placed, and p( x1,x2) the joint probability function for (X1,X2); records indicate that P(0,0)= 0.06 P (2,0)= 0.20 P(1,0)= 0.14 P(2,1)= 0.30 P(1,1)= 0.10 P(2,2) = 0.20 For any given day, the probability of say, two calls and one order is 0.30. Find the correlation coefficient and interpret your results.
The correlation coefficient is:r = -0.332 / (0.683 × 0.566) = -0.994
Correlation Coefficient The correlation coefficient, denoted by r, is a statistical measure of the strength of the relationship between two quantitative variables.
It is always between -1 and 1, where 1 indicates a perfect positive correlation, -1 indicates a perfect negative correlation, and 0 indicates no correlation at all.
In this problem, we have the joint probability function for X1 and X2 as follows:
P(0,0) = 0.06P(2,0) = 0.20P(1,0) = 0.14P(2,1) = 0.30P(1,1) = 0.10P(2,2) = 0.20
The probability of two calls and one order is given as 0.30. Thus:P(2,1) = 0.30
Now, let us find the marginal probabilities of X1 and X2.P(X1 = 0) = P(0, 0) + P(0, 1) + P(0, 2) = 0.06 + 0 + 0 = 0.06P(X1 = 1) = P(1, 0) + P(1, 1) + P(1, 2) = 0.14 + 0.10 + 0 = 0.24P(X1 = 2) = P(2, 0) + P(2, 1) + P(2, 2) = 0.20 + 0.30 + 0.20 = 0.70Similarly,P(X2 = 0) = P(0, 0) + P(1, 0) + P(2, 0) = 0.06 + 0.14 + 0.20 = 0.40P(X2 = 1) = P(0, 1) + P(1, 1) + P(2, 1) = 0 + 0.10 + 0.30 = 0.40P(X2 = 2) = P(0, 2) + P(1, 2) + P(2, 2) = 0 + 0 + 0.20 = 0.20
The expected values of X1 and X2 are:
E(X1) = 0 × 0.06 + 1 × 0.24 + 2 × 0.70 = 1.64
E(X2) = 0 × 0.40 + 1 × 0.40 + 2 × 0.20 = 0.80
Let us now find the expected value of X1X2:
E(X1X2) = 0 × 0.06 + 1 × 0.00 + 2 × 0.30 + 0 × 0.14 + 1 × 0.10 + 2 × 0.20 = 1.14
Thus, the covariance of X1 and X2 is:Cov(X1, X2) = E(X1X2) - E(X1)E(X2) = 1.14 - 1.64(0.80) = -0.332
Finally, the correlation coefficient is given as
r = Cov(X1, X2) / (σ(X1)σ(X2))
Where σ(X1) is the standard deviation of X1 and σ(X2) is the standard deviation of X2.
Let us find the standard deviations of X1 and X2:
Variance of X1:Var(X1) = E(X1^2) - [E(X1)]^2 = 0^2(0.06) + 1^2(0.24) + 2^2(0.70) - (1.64)^2 = 0.4676
Standard deviation of X1:σ(X1) = √Var(X1) = √0.4676 = 0.683
Variance of X2:Var(X2) = E(X2^2) - [E(X2)]^2 = 0^2(0.40) + 1^2(0.40) + 2^2(0.20) - (0.80)^2 = 0.3200
Standard deviation of X2:σ(X2) = √Var(X2) = √0.3200 = 0.566
Thus, the correlation coefficient is:r = -0.332 / (0.683 × 0.566) = -0.994
Therefore, the correlation coefficient is very close to -1.
This means that there is a very strong negative correlation between the number of calls and the number of orders placed. This can be interpreted as follows: As the number of calls increases, the number of orders placed decreases, and vice versa.
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