The conditions on a, b, and c for which the Linear equations has solutions are det(A) ≠ 0, Rank[A|B] = Rank(A).
(a) 3x + 2y - z = a
x + y + 2z = b
5x + 4y + 32 = c
For this system to have solutions, the augmented matrix [A|B] must have a unique solution. This is equivalent to determinant of the coefficient matrix A is non-zero, and the system is consistent (the row rank of the augmented matrix is equal to the row rank of the coefficient matrix). Therefore, the conditions on a, b, and c for this system to have solutions are:
det(A) ≠ 0
Rank[A|B] = Rank(A)
(b) -3x + 2y + 4z = a
-x - 2y + 3z = b
-x - 6y + 232 = 0
For this system to have solutions, the conditions are the same as in (a):
det(A) ≠ 0
Rank[A|B] = Rank(A)
(c) 4x - 2y + 3z = a
2x - 3y – 2z = b
4x - 2y + 32 = c
For this system to have solutions, the conditions are the same as in (a) and (b):
det(A) ≠ 0
Rank[A|B] = Rank(A)
For each system of linear equations to have solutions, the number of equations must be equal to the number of variables, and the determinant of the coefficient matrix must be non-zero.
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Please Help! Will Mark Branliest
Find The Point-Slope And Slope Intercept of the following image.
(Number of Miles depend on the amount of thread thickness (tires))
The point-slope form of data contained in this table is y - 15 = 8.3(x - 7.2).
The slope-intercept form of data contained in this table is y = 8.3x - 10.
How to determine an equation of this line?In Mathematics, the point-slope form of a straight line can be calculated by using this mathematical expression:
y - y₁ = m(x - x₁) or y - y₁ = (y₂ - y₁)/(x₂ - x₁)(x - x₁)
Where:
m represents the slope.x and y are the points.At data point (7.2, 15), a linear equation in standard form for this line can be calculated by using the point-slope form as follows:
y - y₁ = (y₂ - y₁)/(x₂ - x₁)(x - x₁)
y - 15 = (35 - 15)/(4.8 - 7.2)(x - 7.2)
y - 15 = 8.3(x - 7.2)
y - 15 = 8.3x - 25
y = 8.3x - 10
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Someone help me with this math problem
Answer:
1. 26*
2. 17*
3. 23*
You will need a pencil and paper.
Nick is deciding what to wear. He has two baseball hats and wants to wear either a sweatshirt or a sweater. How many different combinations are possible?
4 combinations
5 combinations
6 combinations
7 combinations
The number of different combinations that are possible for Nick, given his baseball hats are A. 4 combinations.
How to find the number of combinations?Nick has two options for hats, and two options for sweatshirts/sweaters. To find the number of combinations, we can multiply the number of options for each item.
There are two baseball hats and two clothes - sweatshirt and sweater.
The number of combinations is:
= 2 (hats) x 2 (sweatshirts/sweaters)
= 4 combinations
In conclusion, as regards the number of combinations that Nick can make, there are four.
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(true/false) suppose the first column of a is three times of the second column. then the homogeneous equation ax
The statement that " Homogeneous equation Ax = 0 has trivial solution if it has at least 1 free variable" is FALSE .
For a system of Linear Equations, a free variable is a variable which can take any value, whereas
the Leading Variable is a variable which is determined by the values of the free variables.
The presence of a free variable means that existence of a non trivial solution to Ax = 0 because a free variable corresponds to a column of A that does not contain a pivot position in the corresponding row of [A|0].
So , the corresponding variable can be set to any non zero value, which gives a non trivial solution to Ax = 0.
Therefore, we can conclude that the homogeneous equation Ax = 0 has the trivial solution if and only if the equation has no free variables.
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The given question is incomplete , the complete question is
The homogeneous equation Ax = 0 has the trivial solution if and only if the equation has at least one free variable. True or False ?
Find the average value of the negative-valued function y=f(x), given that the area of the region bounded by the curve f(x) and x-axis from x=4 to x=10 is 13/5
Given that the area of the region bounded by the curve f(x) and x-axis from x=4 to x=10 is 13/5, the average value of the negative-valued function y=f(x) over the interval [4,10] is -13/30.
The average value of a function f(x) over an interval [a,b] is given by:
average value = (1/(b-a)) * integral from a to b of f(x) dx
In this case, we are given that the function y=f(x) is negative-valued, and the area of the region bounded by the curve and x-axis from x=4 to x=10 is 13/5. This means that the integral of f(x) over the interval [4,10] is equal to -13/5:
[tex]\int\limits^{10}_4 \, f(x) dx[/tex] = -13/5
To find the average value of f(x) over this interval, we divide this integral by the length of the interval:
average value = (1/(10-4)) * [tex]\int\limits^{10}_4 \, f(x) dx[/tex]
= (1/6) * (-13/5)
= -13/30
Therefore, the average value of the negative-valued function y=f(x) over the interval [4,10] is -13/30. This means that if we were to draw a horizontal line at the height of -13/30 over the interval [4,10], the area between this line and the x-axis would be equal to the area of the region bounded by the curve f(x) and x-axis over the same interval.
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Let T:V→V be a linear operator.- Prove that if T is injective and {vi} is linearly independent, then {Tvi} is also linearly independent.- Prove that if T is surjective and {vi} spans V, then {Tvi} also spans V.Proof of 1: Assume that T is injective and {vi} is linearly independent. By definitions,- A linear operation T is called injective if x=0 whenever Tx=0. (i.e. If Tx=0 then x=0)- {vi}ki=1 is linearly independent if the only set of scalars {c1,c2,…,ck} which gives ∑ki=1civi=0 is the zero set c1=c2=⋯=ck=0. \end{itemize}We need to show that {Tvi}ki=1 is linearly independent if the only set of scalars {a1,a2,…,ak} which gives ∑ki=1aiTvi=0 is the zero set a1=a2=⋯=ak=0.
In mathematical terms, if T is injective, then for any vectors u and v in the domain V, T(u) = T(v) only if u = v. This property is also known as one-to-one correspondence.
One important property of linear operators is their infectivity, which means that the function maps distinct vectors in the domain to distinct vectors in the codomain.
To prove that {Tvi} is also linearly independent, we need to show that if a1T(v1) + a2T(v2) + ... + akT(vk) = 0, then the only solution is a1 = a2 = ... = ak = 0. We can do this by assuming that there exist scalars a1, a2, ..., ak that are not all zero, and show that this leads to a contradiction.
Suppose that
=> a1T(v1) + a2T(v2) + ... + akT(vk) = 0,
where not all of the ai's are zero. Then, we can write this as
=> T(a1v1 + a2v2 + ... + akvk) = 0.
Since T is injective, this implies that a1v1 + a2v2 + ... + akvk = 0.
Now, let's move on to the second part of the problem statement, which deals with the surjectivity of the linear operator T.
In other words, if T is surjective, then for any vector w in the codomain V, there exists a vector v in the domain V such that T(v) = w.
If {vi} spans V, then every vector in V can be expressed as a linear combination of the {vi}'s. In other words, for any vector u in V, there exist scalars c1, c2, ..., ck such that
=> u = c1v1 + c2v2 + ... + ckvk.
To prove that {Tvi} also spans V, we need to show that every vector w in V can be expressed as a linear combination of the T(vi)'s.
Then, we can express v as a linear combination of the {vi}'s:
=> v = c1v1 + c2v2 + ... + ckvk.
Applying T to both sides, we get:
w = T(v) = T(c1v1 + c2v2 + ... + ck
W = c1T(v1) + c2T(v2) + ... + ckT(vk)
This shows that w can be expressed as a linear combination of the T(vi)'s, which means that {Tvi} also spans V.
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Which of the equations below could be the equation of this parabola? ASAP
The Equation of Parabola shown in the graph is y=2x².
What is Equation of Parabola?The general equation of a parabola is given by
y = a(x – h)² + k or x = a(y – k)² +h.
Here (h, k) denotes the vertex. y = a(x – h)² + k is the regular form. x = a(y – k)²+h is the sidewise form.
Given:
We have vertex (0, 0).
As, from the Graph the parabola is open up.
So, it is of the form y= ax², a>0.
Thus, the Equation of parabola is y=2x².
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The water level (in feet) at a dock during a certain 24 hour period is approximately given by H(t) = 5 sin (pi/6 (t - 6)) + 19 What is the average water level at the dock between 4 and 19?
The average water level at the dock between 4 and 19 is 17.5 feet, calculated by taking the average of the water level at 4 and 19.
H(4) = 5 sin (pi/6 (4 - 6)) + 19 = 17
H(19) = 5 sin (pi/6 (19 - 6)) + 19 = 18
Average water level = (17 + 18)/2 = 17.5
The given equation is H(t) = 5 sin (pi/6 (t - 6)) + 19. This equation represents the water level at a dock over a 24 hour period, where t represents the time in hours. To calculate the average water level at the dock at 4 and 19, we need to first calculate the water level at 4 and 19. To do this, we substitute 4 and 19 in the equation in place of t. This gives us H(4) = 5 sin (pi/6 (4 - 6)) + 19 = 17 and H(19) = 5 sin (pi/6 (19 - 6)) + 19 = 18. The average water level is then calculated by taking the average of these two values, which is (17 + 18)/2 = 17.5. This is the average water level at the dock between 4 and 19.
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Question 3
Limosine Company P charges a rental fee plus an additional charge per hour. The equation y-50+ 30x shows the total cost, y of renting the limosine, and the x represents the hours. The cost to rent a
limosine from Company R is shown below in the table. Which statement is true?
Company R
1
2
3
4
$100 $125 $150 $175
Time (hours)
Total Cost
A
B
C
D
Company P charges $25 more for its rental fee compared to Company R
Company R charges a higher rental fee, but a cheaper rate of change to rent a limosine
Company P has a lower rate of change than Company R
Company R charges less money for 1 hour of service compared to Company P
22023 Illuminate Education, Inc.
Type here to search
5
$200
Ħi
E
5:38 PM
2/9/202
The correct statements regarding the linear functions are given as follows:
Company P charges $25 more for its rental fee compared to Company R.Company P has a lower rate of change than Company R.How to define the linear function?The slope-intercept definition of a linear function is given as follows:
y = mx + b.
In which:
m is the slope, representing the rate of change.b is the intercept, representing the value of y when x = 0.For Company P, the function is given as follows:
y = 30x + 50.
Meaning that:
The cost per hour is of $30.The rental fee is of $50.From the table, for company R, the function is given as follows:
y = 25x + 75.
Meaning that:
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Crispin wants to calculate the density of an apple. To do this, he measures the apple's mass and its volume. He measures the mass as 202 grams and gets a volume measurement of 270 cm3.
What is the density of the apple, rounded to the correct number of significant figures?
A.
68 grams/cm3
B.
0.75 grams/cm3
C.
0.748 grams/cm3
D.
1.0 grams/cm3
Answer:
B , 0.75 grams/cm3
Step-by-step explanation:
Crispin's mass measurement (202 grams) has three significant figures, but his volume measurement (270 cm3) only has two significant figures. So when one of these factors is divided by the other, the factor with the lowest number of significant figures takes precedence. This means that the density calculation can only have two significant figures.
The density of an object is its mass divided by its volume:
So, 202 grams ÷ 270 cm3 = 0.75 grams/cm3
(I also did this on study island and the other person made me get it wrong so i got the answer)
The selling price of skateboard is $147 the markup is 75% how much did the store pay for the skateboard
Answer: The store paid $84 to purchase the skateboard
Step-by-step explanation:
If the selling price of the skateboard is $147 and the markup is 75%, we can find the cost of the skateboard by working backwards from the selling price.
Let's start by calculating the original cost of the skateboard before the markup was applied. We can do this by dividing the selling price by 1 plus the markup percentage as a decimal:
original cost = selling price / (1 + markup percentage)
Converting the markup percentage to a decimal:
markup percentage = 75% = 0.75
original cost = 147 / (1 + 0.75)
original cost = 147 / 1.75
original cost = 84
Tamika and Li are painting a mural. Tamika has painted 1/3 of the mural and Li has painted 1/6 of it. What part of the mural have they painted in all?
Answer:
Tamika has painted 1/3 of the mural, and Li has painted 1/6 of it. To find the total part of the mural that they have painted together, we need to add these fractions:
1/3 + 1/6
We need to find a common denominator to add these fractions. The smallest common multiple of 3 and 6 is 6. We can convert 1/3 to 2/6 by multiplying the numerator and denominator by 2:
2/6 + 1/6 = 3/6
So, Tamika and Li have painted 3/6 of the mural in all. We can simplify this fraction by dividing both the numerator and denominator by 3:
3/6 ÷ 3/3 = 1/2
Therefore, Tamika and Li have painted 1/2 of the mural in all
PLEASE HELP ASAP DUE SOON!
The domain of the function g(x) is given as follows:
x ≥ -1.
How to obtain the domain of the function?The domain of a function is composed by the set that contains all possible input values for the function.
The square root function is defined only for non-negative values, hence the domain is obtained as follows:
6x + 6 ≥ 0
6x ≥ -6
x ≥ -6/6
x ≥ -1.
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1/2 , 2/4 , and 4/8 are ____.
Answer:
Equivalent fractions.
Step-by-step explanation:
[tex]\dfrac{1}{2}\\\\\dfrac{2}{4}=\dfrac{2 \div 2}{4 \div 2}=\dfrac{1}{2}\\\\\\\dfrac{4}{8}=\dfrac{4 \div 4}{8 \div 4}=\dfrac{1}{2}[/tex]
So, they are equivalent fractions. Equivalent fractions are fractions with different denominators and numerators representing the same part of a whole.
Answer:
thet are equivalent fractions
Step-by-step explanation:
1/2*2/2=2/4,
2/4*2/2=4/8 or
What is the solution set of the quadratic inequality 6x² +1 <0?
Answer: Solution set of the quadratic equation is, Empty set
Step-by-step explanation:
Given the quadratic equation:
Subtraction property of equality states that you subtract the same number to both sides of an equation.
Subtract both sides by 1 we get;
Simplify:
Division property of equality states that you divide the same number to both sides of an equation.
Divide both sides by 6 we get;
Simplify:
For any x in real number there does not exist any number x which satisfy
, therefore, there is no solution for this set of the quadratic inequality or in other word we can say that set of the solution is Empty set.
Please help solve for nmber 4
The requried a measure of the exterior angle ∠4 = 63°.
What is the triangle?The triangle is a geometric shape that includes 3 sides and the sum of the interior angle should not be greater than 180°
Here,
As we know that for a triangle, the sum of the remote interior angle is equal to the measure of the exterior angle.
So, from the figure.
31 + 32 = ∠4
∠4 = 63°
Thus, the requried measure of the exterior angle ∠4 = 63°.
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NO LINKS!! URGENT HELP PLEASE!!
Find the shaded area of each figure, and round your answer to one decimal place if necessary. #4-6
Answer:
38 & 27 & 147
Step-by-step explanation:
4) 38cm²
5*4 = 20cm²
2*9 = 18cm²
5) 27m²
6*6 = 36m²
3*3 = 9m²
6) 147in²
14*14 = 196cm²
7*7 = 49in²
Answer:
4) 38 cm²
5) 27 m²
6) 147 in²
Step-by-step explanation:
To calculate the area of each given figure, subtract the area of the cut-out rectangle (marked in blue on the attached diagram) from the area of the larger rectangle.
[tex]\boxed{\begin{minipage}{5cm}\underline{Area of a rectangle}\\\\$A=w\cdot l$\\\\where:\\ \phantom{ww} $\bullet$ \quad $w$ is the width.\\ \phantom{ww} $\bullet$ \quad $l$ is the length.\\\end{minipage}}[/tex]
Question 4[tex]\begin{aligned}\textsf{Total Area}&=\textsf{Larger rectangle}-\textsf{Cut-out rectangle}\\&=9 \cdot 6- (9-5)\cdot 4\\&=9 \cdot 6- 4\cdot 4\\&=54-16\\&=38\; \sf cm^2\end{aligned}[/tex]
Question 5[tex]\begin{aligned}\textsf{Total Area}&=\textsf{Larger rectangle}-\textsf{Cut-out rectangle}\\&= 6\cdot 6- 3\cdot 3\\&=36-9\\&=27\; \sf m^2\end{aligned}[/tex]
Question 6[tex]\begin{aligned}\textsf{Total Area}&=\textsf{Larger rectangle}-\textsf{Cut-out rectangle}\\&= 14\cdot 14- 7\cdot 7 \\&=196-49\\&=147\; \sf in^2\end{aligned}[/tex]
can i have ten dollahairs
maybe if you ask politely
A factory has three assembly lines, with the first line producing 29% of the product, the second line producing 27% of the product, and the third line producing the remainder. The first line produces defective parts 2% of the time, the second line produces defective parts 4% of the time, and the third line produces defective parts 1% of the time. What is the probability that a defective part made at this factory was made on the first assembly line?
The probability that a defective part made at this factory was made on the first assembly line is 0.299 or 29.9%,
This issue can be resolved using Bayes' theorem. Let L1, L2, and L3 represent the events that the component was produced on the first, second, and third assembly lines, respectively, and let D represent the event that a defective part is produced. Next, given that the part is defective, we want to determine P(L1 | D), which is the likelihood that the component was made on the first production line.
We can determine P(D), the overall likelihood of creating a defective part, using the rule of total probability:
P(D) = P(D | L1) P(L1) + P(D | L2) P(L2) + P(D | L3) P(L3)
= 0.02 × 0.29 + 0.04 × 0.27 + 0.01 × 0.44
= 0.0195
next, we can determine P(L1 | D) using Bayes' theorem:
P(L1 | D) = P(D | L1) P(L1) / P(D)
= 0.02 × 0.29 / 0.0195
= 0.299
As a result, there is a 29.9% chance that a defective component produced at this factory was made on the first assembly line, or about 0.299 of a chance.
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records show that the average number of phone calls received per day is 8.6. find the probability of at most 2 phone calls received in a given day. round your answer to three decimal places, and include all written work usin
The probability of at most 2 phone calls received in a given day is roughly 0.021.
Since we are asked for the likelihood of a specific number of phone calls in a given day and are provided the average daily rate of phone calls, we must use the Poisson distribution to solve this problem.
Let represent the average daily call volume, which is provided as = 8.6. Following that, the likelihood of getting no more than two phone conversations in a day is:
P(X ≤ 2) = e^(-λ) * (λ^0/0!) + e^(-λ) * (λ^1/1!) + e^(-λ) * (λ^2/2!)
= e^(-8.6) * (8.6^0/0!) + e^(-8.6) * (8.6^1/1!) + e^(-8.6) * (8.6^2/2!)
≈ 0.021
Therefore, the probability of receiving no more than 2 phone calls in a given day is roughly 0.021.
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The expected value of an unbiased estimator is equal to the parameter whose value is being estimated. A) True B) False
The expected value of an unbiased estimator is equal to the parameter whose value is being estimated. A) True B) False . the answer is false
Find the surface area AND volume of the following polyhedra. (SHOW ALL
WORK)
8
The simplest way to calculate the surface area of a polyhedron, though, remains to simply sum the areas of the polygons that make up its faces. The surface area of a sphere has a very interesting formula. It depends solely on the radius of the sphere. The surface area of a sphere is equal to 4Π times the square of the radius of the sphere: 4Πr2.
Simplify each expression below by combining like terms. Then match it to it's simplified expression.
3y + 2y + y² +5+y
3y² + 2xy +1+3x+y+2x²
3xy + 5x + 2+ 3y + x +4
4m + 2mn + m² + m +3m²
:: 2z²+9zy + 1
:: 2x² + 3x + 4y + 2xy + 1
:: 12xy + 6
:: 4m² + 4m + 2mn :: y² + 5y + 5
:: 6x + 3y + 3xy +6
:: 4m² + 5m + 2mn
:: y² + 6y + 5
The slope-intercept form of a line is y=2/3x + 2
. Which line is perpendicular and shifted down 8?
The required line is y = -(3/2)x - 6 which is perpendicular and shifted down 8 units.
What are the perpendicular lines?Perpendicular lines are two lines that cross one other and create a 90-degree angle. Two lines are perpendicular if the product of their slopes is -1.
The slope-intercept form of a line is y = (2/3)x + 2.
To find the slope of the perpendicular line, we simply need to negate the reciprocal of the original line's slope.
The slope of the original line is 2/3. The slope of the perpendicular line is therefore -3/2.
To shift the line down 8 units, we simply need to subtract 8 from the y-intercept.
The slope-intercept form of the line perpendicular to y = (2/3)x + 2 and shifted down 8 units is:
y = -(3/2)x + 2 - 8 = -(3/2)x - 6
So the line that is perpendicular and shifted down 8 units is y = -(3/2)x - 6.
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The above PPF of producing mountain bikes to racing bikes.
The factory is currently producing 250 mountain bikes and 100 racing bikes per week (point A). If they wanted to produce 50 more racing bikes per week (point B), what would be there opportunity cost?
A. 100 racing bikes
B. 250 mountain bikes
C. 50 mountain bikes
D. There is no opportunity cost.
They wanted to produce 50 more racing bikes per week (point B), what would be there opportunity cost There is no opportunity cost.
What is meant by speed?
the pace at which an object's location changes in any direction. Speed is defined as the distance traveled divided by the travel time. The speed of sound and light both represent quickness in movement, going, traveling, proceeding, or performance.Full speed ahead. relative quickness in action, advancement, etc. complete, maximum, or ideal rate of motion: Nine seconds are all it takes for the car to reach speed.An object's speed can be determined by calculating the distance it covers in a given amount of time. For instance, an automobile is moving at 70 miles per hour if it covers 70 miles in an hour (miles per hour).The opportunity cost of producing 50 more racing bikes per week (moving from point A to point B) would be the amount of mountain bikes that the factory would have to give up producing in order to allocate resources to producing more racing bikes.
From the PPF, we can see that in order to produce an additional 50 racing bikes per week, the factory would have to decrease its production of mountain bikes by 50 (moving from 250 to 200). Therefore, the opportunity cost of producing 50 more racing bikes per week is 50 mountain bikes (option C).
So the answer is C. 50 mountain bikes.
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A right prism with height $h$ has bases that are regular hexagons with sides of length $12$. A vertex $A$ of the prism and its three adjacent vertices are the vertices of a triangular pyramid. The dihedral angle (the angle between the two planes) formed by the face of the pyramid that lies in a base of the prism and the face of the pyramid that does not contain $A$ measures $60$ degrees. Find $h^2$.
By the face of the pyramid that lies in a base of the prism and the face of the pyramid, h2 = 108°
Let B and C be the vertices adjacent to A on the same base as A, and let D be the last vertex of the triangular pyramid. Then angle CAB = 120. Let x be the foot of the altitude from A to line BC. Then since triangle ABX is a 30-60-90 triangle, AX=6. Since the dihedral angle between triangle ABC and triangle BCD is 60 degree, triangle AXD is a 30-60-90 triangle and
[tex]AD = 6 \sqrt{3} = h[/tex]
Therefore,
[tex] {h}^{2} = 108[/tex]
A triangular prism is a three-sided prism that has three faces joining the corresponding sides of its triangular base, translated copy, and polyhedron construction. If the sides of a right triangular prism are not rectangular, the prism is oblique. Having square sides and equilateral bases, a uniform triangular prism is a right triangle.
It is, in essence, a polyhedron with three faces, all of whose surface normals are in the same plane (which need not be parallel to the base planes), and two of whose faces are parallel. It is a parallelogram that has these three faces. The same triangle can be seen in all cross-sections that are perpendicular to the base faces.
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25. Write an expression to represent the area of the shaded region in simplest form.
5x + 2
x+7
3.x-1
Gina Willson (All Things Alge
I'm sorry, there is no image or diagram provided to identify the shaded region. Could you please provide more information or a visual aid for me?
Find the area of the saved region. The graph depict IQ scores of adults, and those scores are normally distributed with a mean of 100 and a standard deviation of 15. 91
The area of the shaded region is 0.725746882.
What is a cumulative distribution function?The probability distribution of random variables is described using the cumulative distribution function. The probability for a discrete, continuous, or mixed variable may be described using it. The cumulative probability for a random variable is calculated by adding the probability density function.
Given:
The graph depicts the IQ scores of adults, and those scores are normally distributed with a mean of 100 and a standard deviation of 15. 91.
From the diagram,
the critical value is 91.
The formula to calculate the z-score is,
z = (x - u) / s,
Substituting the given values to the formula,
the given values are;
x = critical value = 91
u = mean = 100
s = standard deviation = 15
Thus,
z = (x - u) / s
z = -0.6
P(z > -0.6 ) = 0.725746882
Therefore, the area is 0.725746882.
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The historical returns on a balanced portfolio have had an average return of 5% and a standard deviation of 14%, Assume that returns on this portfolio follow a normal distribution. (You may find it useful to reference the z table.) a. What percentage of returns were greater than 33%? (Round your final answer to 2 decimal places.) Percentage of returns = %
b. What percentage of returns were below -37%? (Round your final answer to 2 decimal places.) Percentage of returns %
The percentages of returns are given as follows:
a. Greater than 33%: 2.28%.
b. Below -37%: 0.13%.
How to obtain probabilities using the normal distribution?The z-score of a measure X of a variable that has mean symbolized by [tex]\mu[/tex] and standard deviation symbolized by [tex]\sigma[/tex] is obtained by the rule presented as follows:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The z-score represents how many standard deviations the measure X is above or below the mean of the distribution, depending if the obtained z-score is positive or negative.Using the z-score table, the p-value associated with the calculated z-score is found, and it represents the percentile of the measure X in the distribution.The parameters for this problem are given as follows:
[tex]\mu = 5, \sigma = 14[/tex]
The proportion above 33% is the one subtracted by the p-value of Z when X = 33, hence:
Z = (33 - 5)/14
Z = 2.
Z = 2 has a p-value of 0.9772.
1 - 0.9772 = 0.0228 = 2.28%.
The proportion below -37% is the p-value of Z when X = -37, hence:
Z = (-37 - 5)/14
Z = -3.
Z = -3 has a p-value of 0.0013.
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Compute the mean and standard deviation of the random variable with the given discrete probability distribution. x P(x)-2 0.12 0.26 3 0.28 4 0.14 7 0.22
Mean of this random variable is 3.14 and standard deviation is 1.82. The mean is calculated by multiplying each value of x by its corresponding probability and then summing the result.
Mean = (-2 * 0.12) + (0 * 0.26) + (3 * 0.28) + (4 * 0.14) + (7 * 0.22)
= 3.14
standard Deviation = [tex]\sqrt{[( (-2)2 * 0.12 ) + (02 * 0.26) + (32 * 0.28) + (42 * 0.14) + (72 * 0.22) - (3.142) ] }[/tex]= 1.82
The mean and standard deviation of a random variable with a given discrete probability distribution can be calculated using the following steps. First, each value of x is multiplied by its corresponding probability and then the results are summed to calculate the mean. For example, for the given discrete probability distribution, the mean is calculated by multiplying -2 by 0.12, 0 by 0.26, 3 by 0.28, 4 by 0.14 and 7 by 0.22 and then summing the results which is 3.14.To calculate the standard deviation, the square of each value of x multiplied by its corresponding probability is summed. Then the square of the mean is subtracted from this result and the square root of the remainder is taken to calculate the standard deviation. For the given probability distribution, the standard deviation is calculated by summing the square of -2 multiplied by 0.12, 0 multiplied by 0.26, 3 multiplied by 0.28, 4 multiplied by 0.14 and 7 multiplied by 0.22. Then the square of the mean which is 3.142 is subtracted from this result and the square root of the remainder is taken which is 1.82.
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