If X and Y are independent variables... prove that mx+y(t) = mx (t)my(t) ✓ use the fact that mx+y(t) = mx(t)my (t) to prove that Var (X+Y) = Var(X) + Var(Y) prove that mx-y(t) = mx (t)my (-t) use the fact that mx_y(t) = mx(t)my (-t) to prove that Var (X + Y) = Var(X) + Var(Y)
Answers
Given that X and Y are independent variables.
Let Mx (t) = E [e^tX]My (t) = E [e^tY]Mx+y (t) = E [e^t (X+Y)]Mx+y (t) = E [e^tX * e^tY] (Since X and Y are independent)Mx+y (t) = E [e^tX] * E [e^tY] (As X and Y are independent, E [X+Y] = E [X] + E [Y])Mx+y (t) = Mx (t) * My (t) ………………
(1)Now, let’s prove that Var (X+Y) = Var (X) + Var (Y)Var (X+Y) = E [(X+Y)^2] – E [X+Y]^2Var (X+Y) = E [X^2 + Y^2 + 2XY] – [E (X) + E (Y)]^2 (Expanding the square of X+Y)Var (X+Y) = E [X^2] + E [Y^2] + 2 E [XY] – [E (X)^2 + E (Y)^2 + 2E (X)E (Y)]Var (X+Y) = (E [X^2] – E [X]^2) + (E [Y^2] – E [Y]^2) + 2 (E [XY] – E [X] E [Y])Var (X+Y) = Var (X) + Var (Y) + 2 Cov (X,Y) ………………
(2)Now, let’s prove that mx-y(t) = mx (t)my (-t)Mx-y (t) = E [e^t (X-Y)] = E [e^tX / e^tY] = E [e^(t (X-Y))] / E [e^tY] ……………… (3) (By the property of division)Multiplying and dividing the numerator of equation (3) by e^tY, we get, Mx-y (t) = E [e^tX + (-Y)] * e^(-tY) / E [e^tY]Mx-y (t) = Mx (t) * My (-t) ………………
(4)Now, let’s prove that Var (X-Y) = Var (X) + Var (Y) – 2 Cov (X,Y)Var (X-Y) = E [(X-Y)^2] – [E (X-Y)]^2Var (X-Y) = E [(X^2 + Y^2 – 2XY)] – [(E (X) – E (Y))]^2Expanding the square in the second term, we get, Var (X-Y) = E [X^2 + Y^2 – 2XY] – E [X]^2 – E [Y]^2 + 2E [X] E [Y]Var (X-Y) = (E [X^2] – E [X]^2) + (E [Y^2] – E [Y]^2) – 2 (E [XY] – E [X] E [Y])Var (X-Y) = Var (X) + Var (Y) – 2 Cov (X,Y) ………………
(5) From equations (2) and (5), we can write, Var (X+Y) + Var (X-Y) = 2 (Var (X) + Var (Y)) Therefore, Var (X+Y) = Var (X) + Var (Y)
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Related Questions
A cosine function has a period of 4, a maximum value of 20, and a minimum value of 0. The function is a not a reflection of its parent function over the x-axis.
Which function could be the function described?
Answers
Answer:
Step-by-step explanation:
You'll need to find a way to share the graph or graphs associated with this problem.
The period is 4. The relationship between period and b (the coefficient of the independent variable of the cosine function) is (period) =(2π.b). Given that the period here is 4,
2π 2π
4 = ---------, and so b = ------- = π/2.
4
So now our y = a*cos (bx + c) becomes
π
y = 10*cos (------x + 0 + 10
2
The horizontal center line of this cosine curve is y = 10. The maximum value of the curve is 20, which is 10 + 10, and the minimum value is 0, which is 10 - 10.
The function that describes the given situation is \(\rm y = 10\;cos\left(\dfrac{\pi}{2}x\right) + 10\) and this can be determined by using the properties of the trigonometric function.
Given :
A cosine function has a period of 4, a maximum value of 20, and a minimum value of 0. The function is not a reflection of its parent function over the x-axis.The following steps can be used in order to determine the function could be the function described:
Step 1 - According to the given data, the cosine function has a period of 4.
Step 2 - Now, the independent variable of the cosine function is given by:
\(\rm 4b = 2\pi\)
\(\rm b =\dfrac{\pi}{2}\)
Step 3 - The maximum value of the cosine function is 1 and the minimum value of the cosine function is -1.
Step 4 - From the above steps, the function that described the given situation is given by:
\(\rm y = 10\;cos\left(\dfrac{\pi}{2}x\right) + 10\)
Therefore, the correct option is C).
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how many prime numbers are there between 30 and 50
Answers
A prime number is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. To find out how many prime numbers there are between 30 and 50, we need to check each of the numbers from 30 to 50 to see if it is a prime number.
A prime number is a number that is only divisible by 1 and itself. The prime numbers between 30 and 50 are: 31, 37, 41, 43, and 47.
Therefore, there are 5 prime numbers between 30 and 50.
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. Using 1 2 3 4 5 ,
write at least ten different mathematical expressions. Use at least one
of the numbers as an exponent. Include the mathematical operations
of addition and multiplication, and use parentheses when writing the
expressions. Each of the 10 expressions should simplify to a different
value. The write at least two pairs of expressions that have the same
value
Answers
Answer:
Influence ideas and feelings about sexuality, your religion
The city of London, England, has an
elevation of 11 meters.
Which of these describes the elevation
of London?
below sea level
at sea level
above sea level
Answers
Answer:
above sea level
Step-by-step explanation:
HELP PLEASE ILL GIVE 50 POINTS PLEASE DUE AT 11:59 WHERE I LIVE I HAVE AN HOUR!!!
i need a conclusion paragraph to the question - Was Abby Sunderland too young to sail solo? - SAYING THAT SHE WAS!
pleaseee the best conclusion will get brainliest!!!!!!!!!!
i am aware the subject is math accident
Answers
Answer:
Step-by-step explanation:
In conclusion, it can be argued that Abby Sunderland was too young to sail solo. Despite her ambitious and determined nature, her lack of experience and maturity put her at a significant risk during her solo sailing journey. The fact that she had to be rescued twice during her attempt highlights the dangers of attempting such a feat at such a young age. While it is admirable to have the courage to pursue one's dreams, it is important to consider the potential consequences and weigh them against the potential benefits. In this case, it can be argued that the risks far outweighed the potential rewards.
You have a circular loop of wire in the plane of the page with an initial radius of 0.40 m which expands to a radius of 1.00 m. It sits in a constant magnetic field B = 24 mT pointing into the page. Assume the transformation occurs over 1.0 second and no part of the wire exits the field. Also assume an internal resistance of 30 Ω. What average current is produced within the loop and in which direction? Express your answer with the appropriate units. Enter positive value if the current is clockwise and negative value if the current is counterclockwise. My INCORRECT work: emf = -BAcos(theta)/dt emf = -B*1*(dA/dt) emf = -B*pi*(2*(expansion rate/1)^2*t+2*r0*(expansion rate/1) Then V=IR so emf=IR so I=emf/R I = -[B*pi*(2*(expansion rate/1)^2*t+2*r0*(expansion rate/1)]/R I = -[24x10^-3*pi*(2*.6^2*1+2*.4*.6)]/30 I ~ -3.015928947x10^-3 I ~ -3.0x10^-3 Which is wrong.
Answers
In the given scenario, the average current produced within the loop is approximately 2.13 A.
We can begin by computing the change in magnetic flux across the loop as it expands to determine the average current generated within the loop.
The following equation provides the magnetic flux across a loop:
Φ = B * A * cos(θ)
ΔΦ = B * ΔA
ΔA = A₂ - A₁ = π * (1.00 m)² - π * (0.40 m)² = π * (1.00² - 0.40²) = π * (1.00 + 0.40)(1.00 - 0.40) = π * (1.40)(0.60) = 0.84π m²
So,
ΔΦ = B * ΔA = (24 mT) * (0.84π m²) = 20.25π m²·T
emf = ΔΦ / Δt = (20.25π m²·T) / (1.0 s) = 20.25π V
As:
emf = I * R
So, again
I = emf / R = (20.25π V) / (30 Ω) ≈ 2.13 A
Thus, the average current produced within the loop is approximately 2.13 A.
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A trapezoid has consecutive sides of length 8, 10, and 12 in that order. Find the largest possible area of such a trapezoid rounded down to the nearest integer.
Answers
The largest possible area of such a trapezoid rounded down to the nearest integer is 108.
The two parallel facets are the bases, and top, as constantly, is the perpendicular distance from one base to the opposite. The place of this parallelogram is its peak (half-peak of the trapezoid instances its base
Area of a trapezoid is = 1/2 * sum of the two parallel side * height
= 1/2 * (12 + 10 ) * 8
= 88
Area of a trapezoid is = 1/2 * sum of the two parallel side * height
= 1/2 * (12 + 8) * 10
= 100
Area of a trapezoid is = 1/2 * sum of the two parallel side * height
= 1/2 * (8+ 10 ) * 12
= 108
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triangle jkl is formed by plotting three points, as shown below. What is the length of side jk
Answers
Answer: 5 wil be your answer
Step-by-step explanation:
just did it
Suppose $726.56 is deposited at the end of every six months into an account earning 6.45% compounded semi-annually. If the balance in the account four years after the last deposit is to be $31 300.00, how many deposits are needed? (This question asks for 'n')
Answers
We need approximately 10 deposits to reach a balance of $31,300 four years after the last deposit which is compounded semi-annually.
To solve this problem, we can use the formula for the future value of an annuity:
\(FV = P * ((1 + r)^n - 1) / r\)
Where:
FV is the future value of the annuity
P is the periodic payment or deposit amount
r is the interest rate per period
n is the number of periods
In this case, the deposit amount is $726.56, the interest rate is 6.45% compounded semi-annually, and the future value is $31,300. We need to find the number of deposits (n).
We can rearrange the formula and solve for n:
n = log((FV * r) / (P * r + FV)) / log(1 + r)
Substituting the given values:
n = log((31,300 * 0.03225) / (726.56 * 0.03225 + 31,300)) / log(1 + 0.03225)
Using a calculator or software, we find that n ≈ 9.989.
Therefore, we need approximately 10 deposits to reach a balance of $31,300 four years after the last deposit.
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How many sides does a regular polygon with 45° exterior angles have?
Numerical answer only.
Answers
Answer:
8 sides (octagon)
Step-by-step explanation:
sum of exterior angles is always 360º
360 / 45 = 8 sides
Answer:
8 sides
Step-by-step explanation:
If all the angles of the polygon are the same, divide the angle by 360 and u get the answer.
quadrilateral abcd is inscribed in a circle. find the measure of each of the angles of the quadrilateral. show your work. hint: see pgs. 424 and 246-249 in your textbook for some examples. also see lesson 3.06 learn > a closer look: explore relationships between an inscribed angle and its intercepted arc and lesson 3.11 learn > a closer look: solve problems involving inscribed quadrilaterals.
Answers
I'll explain the concept and steps to find the measure of each angle in an inscribed quadrilateral.
In an inscribed quadrilateral (a quadrilateral with its vertices on a circle), the opposite angles are supplementary. In other words, the sum of opposite angles is 180 degrees. This relationship between angles can help us find the measure of each angle in the quadrilateral.
Let's denote the angles of quadrilateral ABCD as follows:
∠A, ∠B, ∠C, and ∠D.
Since opposite angles are supplementary:
∠A + ∠C = 180°
∠B + ∠D = 180°
To find the measure of a each angle, you'll need to be given at least two angle measures or additional information about the relationships between the angles. Without specific information, we can only provide the general relationships between the angles in an inscribed quadrilateral.
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The total amount of kinetic and potential energy in a system is called
Answers
The total amount of kinetic and potential energy in a system is called mechanical energy.
Mechanical energy is the sum of kinetic energy and potential energy within a system. Kinetic energy (KE) is the energy possessed by an object due to its motion, and it is calculated using the formula KE = (1/2)mv², where m is the mass of the object and v is its velocity. Potential energy (PE) is the energy possessed by an object due to its position or condition, and it can be gravitational potential energy, elastic potential energy, or other forms. The total mechanical energy (ME) is obtained by adding the kinetic and potential energy together, so ME = KE + PE.
In summary, the total amount of kinetic and potential energy in a system is referred to as mechanical energy. This energy accounts for both the motion and position-related energies within the system. The calculation of mechanical energy involves summing the kinetic energy, which depends on the mass and velocity of the objects, with the potential energy, which can arise from factors such as gravity or elasticity. Understanding and analyzing the mechanical energy in a system can provide insights into its overall energy distribution and behavior.
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The equation y = 2.50x determines the total cost, y, of buying x carnival tickets. Use the drop-down menu to complete the sentence. According to the equation, 15 carnival tickets would cost .
Answers
Answer:
$37.50
Step-by-step explanation:
In this problem, all we need to do is enter the given value into the equation. In this case, \(x\) carnival tickets = 15. So,
\(y=2.50(15)\)
\(y=37.50\)
The total cost of 15 carnival tickets is $37.50.
Hope this helps!
Answer:
37.50
Step-by-step explanation:
Drag the circle to the correct "GCF" of each polynomial.
Answers
The first expression is
8y^4 + 4y^3 + 16y - 8y
The greatest common factor is the highest factor common to each all the terms. For this expression, the GCF is
4 and y because 4 is the highest number that all the terms can divide without a fraction and a single y is common to all that terms. Thus,
GCF = 4y
The second expression is
3x^4 + x^3 - 3 + 3x
There is no number or term that is common to all the terms in the expression. Thus
No GCF
Let A = {2,4,6,8,10,12} B = {3,6,9,12,15,18} C = {0,6,12,18} Find C-A. none of the choices {2,3,4,6,8,9,10,12} O {2,4,8,10) {0,18}
Answers
the correct choice is {0, 18}. These elements are unique to set C and do not appear in set A.
To find the set difference C - A, we need to remove all elements from A that are also present in C. Let's examine the sets:
C = {0, 6, 12, 18}
A = {2, 4, 6, 8, 10, 12}
We compare each element of A with the elements of C. If an element from A is found in C, we exclude it from the result. After the comparison, we find that the elements 2, 4, 8, 10 are not present in C.
Thus, the set difference C - A is {0, 18}, as these are the elements that remain in C after removing the common elements with A.
Therefore, the correct choice is {0, 18}. These elements are unique to set C and do not appear in set A.
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Evaluate the expression when b=2
12-b
Answers
Answer:
10 \(\huge\checkmark\)
Step-by-step explanation:
Hi there, hope you are having a nice day!
All we should do is plug in the value of b:
12-2
10 (Answer)
Hope you find it helpful.
Feel free to ask if you have any questions.
\(\bf{-MistySparkles^**^*\)
Answer:
10
Step-by-step explanation:
If b = 2,
12 - b equals 12 - 2,
thus the result is 10.
Need help with this question ASAP PLEAS
Answers
Answer:
5.36
Step-by-step explanation:
.67 x 8 = 5.36
Helping in the name of Jesus.
The diameter of the cylinder barrel was 20 in. When the barrel was lifted away it left an imprint of a circle on the ground. What is the area of the circle?
Answers
The area of the circle is 1256.64 square inches.
What is a Circle ?All points on a plane that are at a specific distance from a specific point, the center, form a circle. In other words, it is the curve that a moving point in a plane draws to keep its distance from a specific point constant.
A closed 2D form that isn't a polygon is a circle. It has one face that curves.If two circles have the same radius, they are said to be congruent.Equal chords are always located equally from the circle's center.The center of the circle is the perpendicular bisector of a chord.The line joining the points of two overlapping circles will be perpendicular to the line joining the points of the circles' centers.The tangents drawn at the diameter's endpoints are perpendicular to one another.One of the most fundamental curvilinear geometric forms, a cylinder has historically been a three-dimensional solid. It is regarded as a prism with a circle as its basis in basic geometry. In several contemporary fields of geometry and topology, a cylinder can alternatively be characterized as an infinitely curved surface.
The diameter of the cylindrical barrel is 20 in
So the diameter of the circular base is 20 in = 10 inches in radius.
Area of the circle = \(\pi\ r^{2}\) = π*20*20 = 1256.64 square inches.
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now consider that the penetrance of the disease is 75%. what is the probability that individual ii-2 exhibits this disease given this penetrance?
Answers
The probability that individual ii-2 exhibits the disease given a 75% penetrance is 0.75 or 75%.
Penetrance refers to the probability of an individual expressing a particular trait or disease given their genetic makeup. In this case, the penetrance of the disease is 75%, which means that there is a 75% chance that an individual with the genetic mutation for the disease will actually exhibit the disease. Therefore, the probability that individual ii-2 exhibits the disease given this penetrance is also 0.75 or 75%.
1. Penetrance is the probability that an individual with a specific genotype will express the associated phenotype or disease.
2. In this case, the penetrance of the disease is 75%, which means there is a 75% chance that an individual with the disease-associated genotype will exhibit the disease.
3. Given this penetrance, the probability that individual ii-2 exhibits the disease is 0.75 or 75%.
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the total costs of ears of corn at the grocery depends upon how many ears of corn you buy . five ears cost $1.50 and 8 ears cost $2.40 . what is the constant rate of change \
Answers
The constant rate of change for an ear of corn at the grocery store is $0.30.
What is the constant rate of change?The constant rate of change describes the slope of a linear relationship.
The constant rate of change is the constant ratio of the output to the input in the function.
We can compute the constant rate of change by forming a set of simultaneous equations from the given variables.
Total Costs of Ears of Corn:
5 ears = $1.50 ... Equation 1 = 5c = $1.50
8 ears = $2.40 ... Equation 2 = 8c = $2.40
Subtract Equation 1 from Equation 2:
Difference = 3c = $0.90 ($2.40 - $1.50)
c = $0.30 ($0.90/3)
Constant rate of change = $0.30
Thus, the constant rate of change implies that each ear of corn at the grocery costs $0.30.
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I was supposed to simplify - (-a - 1) - (a - 1), I got 2a + 2, and got the answer wrong.
Answers
Simplify:
\(\begin{gathered} (-a-1)-(a-1)= \\ -a-1-a+1= \\ -2a \end{gathered}\)Answer: -2a
what is the term for the value that occurs most often in a series of numbers?
Answers
The term for the value that occurs most often in a series of numbers is called the mode.
The mode is one of the three main measures of central tendency, along with the mean and the median. It is a useful descriptive statistic that can provide insights into the characteristics of a dataset.
To find the mode of a set of data, you first need to arrange the data in order, either in increasing or decreasing order. Then, you simply identify the most frequent data point, which is the mode. In some cases, there may be more than one mode if multiple data points occur with the same maximum frequency.
The mode is particularly useful when dealing with categorical or nominal data, where there are distinct categories or values that cannot be ordered in a meaningful way. For example, the mode can help identify the most popular color among a group of people or the most common type of car on a given street. It can also be used for continuous data, although it may be less useful in this case than the mean or median.
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Find the DISTANCE of line segment AB with point A
Answers
Answer:
34
Step-by-step explanation:
x-y
How do you write 0.000003939 in scientific notation?
Answers
That should be right.
information retrieval tools include database management systems, reporting tools, and ________________ tools.
Answers
Information retrieval tools include database management systems, reporting tools, and search engine tools.
Search tools employ various algorithms and techniques to index and search through vast amounts of data. They create indexes or catalogs of the content, allowing users to search for specific keywords or criteria. These tools utilize techniques like keyword matching, relevance ranking, and advanced search operators to refine search results and provide relevant information to users.
Indexing tools, on the other hand, focus on creating structured representations of data for efficient retrieval. They organize and categorize data based on predefined criteria, such as document types, dates, or categories. These tools help users navigate and explore information repositories more easily.
Overall, search and indexing tools are essential components of information retrieval systems. They enhance the accessibility and usability of data by enabling users to locate, retrieve, and analyze information quickly and accurately.
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Write the equation of a line with a slope of -2 and a y-intercept of -5
Answers
Answer:
y = -2x - 5
Step-by-step explanation:
y = is a standard for any equation, as is the use of x as a variable, unless something else is specified.
-2x is the way to denote a slope of -2, and -5 gives you the y-intercept of 5. One point on this line would be (0, -5).
Ava’s car will hold 26 cartons of books. What is the least number of trips she must make in order to deliver 250 cartons?
Answers
The least number of trips that Ava must make in order to deliver 250 cartons of books is 10 trips.
The least number of trips that Ava must make in order to deliver 250 cartons of books can be found by dividing the total number of cartons by the number of cartons that her car can hold per trip.
This can be written as:
Least number of trips = Total number of cartons / Number of cartons per trip
Substituting the given values into the equation, we get:
Least number of trips = 250 cartons / 26 cartons
Least number of trips = 9.62
Since Ava cannot make a fraction of a trip, we need to round up to the nearest whole number. Therefore, the least number of trips that Ava must make in order to deliver 250 cartons of books is 10.
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help please and thank you<3
Answers
Answer:
D. 3 parts
Step-by-step explanation:
10
c(n) = -------; c(n) = 1,300,000
3n
10
(3n)1,300,000 = ------
3n(3n)
3,900,000n = 10
÷3,900,000 ÷3,900,000
---------------- --------------------
n = 0.0000025641 or 2.5641 × 10⁻⁶
n = 2.5641 or rounded to 3
I really hope this helps!
Describe the following characteristics of the graph of each
parent function: domain, range, intercepts, symmetry,
continuity, end behavior, and intervals on which the graph is
increasing/decreasing. (Example 1)
2. f(x) = /1
5. f(x) = c 6. f(x) = x
1. f(x) = [x]
4. f(x) = x4
3. f(x) = x³
Answers
Step-by-step explanation:
-by-step explanation:
y=\frac{2}{x+1}-3y=
x+1
2
−3
asymptotes:
x+1 = 0
x = -1
y = -3
asymptotes: x = -1, y = -3
x int: (-\frac{1}{3}, 0)(−
3
1
,0)
y int: (0, -1)
D: (-infinity, -1) (-1, infinity) x cannot be -1
R: (-infinity, -3) (-3, infinity) y cannot be -3
Find the inverse of each function.
Answers
The inverse of the function (1/2)^x/3
Given function,
y = 3\(log_{1/2}\) x
Now,
y = 3\(log_{1/2}\) x
Let y = f(x)
then,
x = \(f^{-1} (y)\)
Now put \(f^{-1} (y)\) in the place of x,
y = 3\(log_{1/2}\) \(f^{-1} (y)\)
Simplifying,
y/3 = \(log_{1/2} f^{-1} (y)\)
\(f^{-1} (y) = 1/2^{y/3}\\\)
Replace y variable with x,
\(f^{-1} (x)\) = \((1/2)^{x/3}\)
Hence the inverse of function is \((1/2)^{x/3}\) .
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