To show that Sk - ku is a martingale, we need to show that it satisfies the two properties of a martingale:
Sk - ku is integrable for all k.
For any k, E[Sk+1 - ku+1 | E0,...,Ek] = Sk - ku.
First, we show that Sk - ku is integrable for all k:
E[|Sk - ku|] = E[|(X1 + ... + Xk) - k|]
= E[|X1 + ... + Xk| - k] (using linearity of expectation)
≤ E[|X1| + ... + |Xk|] + k (by triangle inequality)
= kE[|X1|] + ... + kE[|Xk|] + k (using linearity of expectation)
= kE[|X1|] + ... + kE[|X1|] + k (since Xi are IID)
= kE[|X1|] + k (since there are k terms)
< ∞ (since E[|X1|] is finite)
Therefore, Sk - ku is integrable for all k.
Next, we show that for any k, E[Sk+1 - ku+1 | E0,...,Ek] = Sk - ku:
E[Sk+1 - ku+1 | E0,...,Ek] = E[(X1 + ... + Xk+1) - (k+1) | E0,...,Ek]
= E[(X1 + ... + Xk) + Xk+1 - (k+1) | E0,...,Ek]
= E[(X1 + ... + Xk) - k | E0,...,Ek] + E[Xk+1 | E0,...,Ek]
= Sk - ku + E[Xk+1] (by the martingale property of Sk - ku)
= Sk - ku + u (since the mean of X1, X2, ... is u)
Therefore, Sk - ku is a martingale.
Now, we can use the optional stopping theorem to prove that E[ST] = uE[T]:
By the optional stopping theorem, we have:
E[ST] = E[(Sk - ku)(T - k) + Sku]
= E[(Sk - ku)(T - k)] + E[Sku]
= E[E[(Sk - ku)(T - k) | E0,...,Ek]] + E[Sku] (using law of total expectation)
= E[(Sk - ku)E[T - k | E0,...,Ek]] + E[Sku] (using linearity of expectation)
= E[(Sk - ku)(E[T | E0,...,Ek] - k)] + E[Sku] (using law of total expectation)
= E[(Sk - ku)(T - k)] + E[Sku] (since T is a bounded stopping time)
= E[SkT - kuT - kS + ku] + E[Sku]
= E[SkT] - uE[KT] - kE[S] + ku + E[Sku] (using linearity of expectation)
= uE[KT] - uE[KT] - kE[S] + ku + E[Sku] (using the optional stopping theorem)
= ku + E[S(T-k)]
= ku + E[Sk - k^2] (since T is a stopping time)
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on the map , a line of length 5cm represents 40km, how many kilometers would a line of length 7 1/2 cm represents
Lois is planning to retire from her job. She has been working there for 27 years. If she retires this year, her monthly pension will be calculated with a flat benefit formula using the flat amount of $52 for each year of
service. She was informed that if she works another 4 years that the flat amount would increase to $60 for each year of service. How much more would her pension be if she waited the 4 years?
If Lois is retiring from her job after 27 years, then if she waited for 4 years, her pension will be $38 higher than if she retires this year.
If Lois retires this year, the flat-amount is $52, So, her monthly pension can be calculated as follows;
⇒ Monthly pension = (52 × 27)/12,
⇒ Monthly pension = $117,
If Lois works for another 4 years, the flat amount used to calculate her monthly pension will increase to $60 for each year of service.
So, her monthly pension will be calculated as follows:
⇒ Monthly pension = (60 × 31)/12,
⇒ Monthly pension = $155,
So, The difference in monthly pension between retiring this year and retiring after working another 4 years can be calculated as follows:
⇒ Difference in monthly pension = $155 - $117,
⇒ Difference in monthly pension = $38,
Therefore, if Lois waits another 4 years before retiring, her monthly pension will be $38 higher than if she retires this year.
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Triangle LMN is an obtuse triangle with l = 6, n = 8.3, and m∠M = 132°. What is the area of ΔLMN?
The area of triangle LMN is 49.61 square units.
We have,
To find the area of triangle LMN.
Area = (1/2) x base x height
where the base and height are the sides of the triangle perpendicular to each other.
First, we need to find the length of side L.
We can use the Law of Cosines.
L² = M² + N² - 2MNcos(132°)
L² = 6² + 8.3² - 268.3cos(132°)
L² = 36 + 68.89 + 99.36
L² = 204.25
L = √204.25
L = 14.28
Now that we know the length of the base of the triangle, we can find the height by dropping a perpendicular from vertex M to side LN.
Since triangle LMN is obtuse, the perpendicular will fall outside of the triangle.
Let's call the foot of the perpendicular point P.
Since M∠M is 132°, we can find the measure of the acute angle ∠LMP:
M∠LMP = 180° - M∠M
M∠LMP = 180° - 132°
M∠LMP = 48°
We can use trigonometry to find the length of MP:
tan(48°) = h/6
h = 6 x tan(48°)
h = 6.93
The area of triangle LMN is:
Area = (1/2) x base x height
Area = (1/2) x 14.28 x 6.93
Area ≈ 49.61
Thus,
The area of triangle LMN is 49.61 square units.
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Need problem please please
The formula for the nth term of the geometric sequence is given as follows:
[tex]a_n = 7(-3)^{n - 1}[/tex]
What is a geometric sequence?A geometric sequence is a sequence of numbers where each term is obtained by multiplying the previous term by a fixed number called the common ratio q.
The formula for the nth term of the sequence is given as follows:
[tex]a_n = a_1q^{n-1}[/tex]
The first term of the sequence is given as follows:
[tex]a_1 = 7[/tex]
Each term is the previous term multiplied by -3, hence the common ratio is given as follows:
q = -3.
Thus the formula for the sequence is given as follows:
[tex]a_n = 7(-3)^{n - 1}[/tex]
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Every number has only one quality: its distance from zero.
O A. True
OB. False
Answer:
True
Step-by-step explanation:
Answer:
False
Step-by-step explanation:
Every number has multiple qualities beyond its distance from zero. Some of the qualities of a number include its sign (positive or negative), its magnitude (Absolute Value), its parity (whether it is even or odd), and its relationship to other numbers (greater than, less than, or equal to). Additionally, numbers can have properties such as being prime or composite, rational or irrational, and real or complex. Therefore, a number is characterized by more than just its distance from zero.
Need help easy please give right answer
Answer: x < 5, see attached
Step-by-step explanation:
To solve for x, we will isolate the given variable.
Given:
[tex]\displaystyle 1 > \frac{x}{5}[/tex]
Multiply both sides of the equation by 5:
[tex]\displaystyle5 > x[/tex]
Flip equation:
[tex]\displaystyle x < 5[/tex]
To graph, we will first plot a point on 5 with an open circle. Then, we will shade to the left since x is less than 5. See attached.
(1 point) (a) Find a vector parametric equation for the part of the saddle = z = x y inside the cylinder 2+2=9 x 2 + y 2 = 9 .
Find the solution set for the following equation. - square root x =5
The solution set for the given equation is 25.
What is a square root function?In Mathematics and Geometry, a square root function is a type of function that typically has this form f(x) = √x, which basically represent the parent square root function i.e f(x) = √x.
Next, we would determine the solution set for the given equation by translating the word problem into an equation and then divide both sides of the square root function as follows;
-√x = 5
√x = -5
By taking the square of both both sides of the square root function, we have:
x = (-5)²
x = 25
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A set of data items is normally distributed with a mean of 500. Find the data item in this distribution that corresponds to the given z-score. z = -3, if the standard deviation is 80.
Answer:
260
Step-by-step explanation:
The z score for a normal distribution is given by the formula
[tex]z = \dfrac{X-\mu}{\sigma}[/tex]
where
X is the data point, μ is the mean and σ is the standard deviation
Given
Mean μ = 500
Standard deviation σ = 80
z-score = -3
and plugging in these values for z-score formula we get
[tex]-3 = \dfrac{X -500}{80}[/tex]
==>
-240 = X - 500
-240 + 500 = X
260 = X
or
X = 260
Luciana bought a total of 7 textbooks at the bookstore. She bought four more textbooks than journals. How many journals did she buy
what is the value of t
If m varies jointly as p and the square root of w and m=21 when p=5 and w=36, find m when p=9 and w=225
When p = 9 and w = 225, the value of m is 94.5
How to find the value of m?We know that m varies jointly as p and the square root of w, then we can write.
m = k*p*√w
Where k is a constant.
We know that m=21 when p=5 and w=36, replacing that we will get:
21 = k*5*√46
21 = k*30
21/30 = k
7/10 = k
Then the equation is:
m = (7/10)*p*√w
Then when p = 9 and w = 225 we will have:
m = (7/10)*9*√225
m = 94.5
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The market mechanism might leave some people with too little income and others with too much. The government uses taxes and transfers to redistribute income more fairly.
Income Transfers are payments to individuals for which no current goods or services are exchanged (e.g., Social Security, welfare, unemployment benefits).
Program Recipient Group Number of Recipients Value of Transfers
Social Security Retired and disabled workers 70 million $ 1,156 billion
Medicare Individuals over age 65 61 million $ 820 billion
Medicaid Medically needy individuals 77 million $ 650 billion
Unemployment compensation Unemployed workers 18 million $ 150 billion
Food stamps Low-income households 43 million $ 105 billion
Earned Income Tax Credit Low-wage workers 25 million $ 62 billion
Temporary Aid to Needy Families Low-income families 3 million $ 30 billion
Calculate the average benefit a Medicaid recipient receives.
Instructions: Round your response to the nearest dollar.The market mechanism might leave some people with too little income and others with too much. The government uses taxes and transfers to redistribute income more fairly.
Income Transfers are payments to individuals for which no current goods or services are exchanged (e.g., Social Security, welfare, unemployment benefits).
Program Recipient Group Number of Recipients Value of Transfers
Social Security Retired and disabled workers 70 million $ 1,156 billion
Medicare Individuals over age 65 61 million $ 820 billion
Medicaid Medically needy individuals 77 million $ 650 billion
Unemployment compensation Unemployed workers 18 million $ 150 billion
Food stamps Low-income households 43 million $ 105 billion
Earned Income Tax Credit Low-wage workers 25 million $ 62 billion
Temporary Aid to Needy Families Low-income families 3 million $ 30 billion
Calculate the average benefit a Medicaid recipient receives.
Instructions: Round your response to the nearest dollar.
Taxes and transfers are used by the government to more equitably share revenue. A Medicaid participant typically receives $1500 in benefits.
We know that,
Medicare is a health insurance program for seniors, disabled individuals under 65, and dialysis patients. Medicaid is a program that provides financial aid for low-income patients' medical costs.
The federal government and the states share funding for the Medicaid program. The amount that the federal government reimburses states for program costs is known as the Federal Medical Assistance Percentage, or FMAP (FMAP).
Spending on Medicaid is mostly targeted at the elderly and those with disabilities. Despite accounting for over two-thirds of Medicaid spending, seniors and individuals with disabilities make up one in four beneficiaries, which reflects the high per-enrollee expenses for both acute and long-term care.
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(Time limit) Give me the domain and range and tell me if the graph is a function or not.
Yes, The given graph is shows a function.
And, Range = {y | y ≥ - 2}
Domain = (- ∞ , ∞)
We have to given that;
the graph is shown in image.
Now, We can check the graph is a function or not by vertical line test.
Hence, By vertical line test it's shows the function as it cut at one point for each value of y.
Thus, The given graph is shows a function.
Since, The set of all inputs of the function is called Domain of set.
Hence, We get;
Domain = (- ∞ , ∞)
And, The set of all outputs of the function is called range of set.
Hence, We get;
Range = {y | y ≥ - 2}
Thus, Yes, The given graph is shows a function.
And, Range = {y | y ≥ - 2}
Domain = (- ∞ , ∞)
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How many ways to paint a house with 3 different colors from a pool of 12 distinct colors
Krell Industries has a share price of $22.08 today. If Krell is expected to pay a dividend of $0.98 this year, and its stock price is expected to grow to $24.98 at the end of the year, what is Krell's dividend yield and equity cost of capital?
what is Krell's equity cost of capital, or expected total return ?
Krell's equity cost of capital, or expected total return, is 5.57%.
We have,
The dividend yield is the annual dividend divided by the current share price:
Dividend yield
= Annual dividend / Share price
= $0.98 / $22.08 = 0.0444 = 4.44%
The equity cost of capital is the expected total return on the stock, which includes both the dividend yield and the expected capital appreciation:
Equity cost of capital = Dividend yield + Capital Appreciation
= 4.44% + [(24.98 - 22.08) / 22.08]
= 4.44% + 0.1314
= 5.57%
Therefore,
Krell's equity cost of capital, or expected total return, is 5.57%.
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apllo traveld 48 leauges in 3 day then he tripled his rate how long would it take him to travel 336 leagues at this new speed
WHAT IS THE RANGE OF 28 42 36 23 14 47 40
Answer:
33
Step-by-step explanation:
To solve for the range, simply subtract the largest-value term from the smallest-value term. First, set all numbers from least to greatest:
14 , 23 , 28 , 36 , 40 , 42 , 47
The smallest term is 14.
The largest term is 47.
Subtract 47 with 14:
[tex]47 - 14 = 33[/tex]
The given range value will be 33.
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A college student takes the same number of credits each semester. Before beginning college, the student had some credits earned while in high school. After 2 semesters, the student had 39 credits, and after 5 semesters, the student had 75 credits.
If C(t) is the number of credits after t semesters, write an equation that represents this situation.
Provide your answer below:
The equation that represents the number of credits the student has earned after t semesters is C(t) = 12t + 15.
Let x be the number of credits the student had before starting college, and let y be the number of credits the student takes each semester. We can use this information to write an equation that represents the number of credits the student has earned after t semesters.
After 2 semesters, the student had 39 credits, which means the student earned 2y + x credits in those two semesters. Similarly, after 5 semesters, the student had 75 credits, which means the student earned 5y + x credits in those five semesters.
Using this information, we can write the following equation:
C(t) = yt + x
where C(t) represents the number of credits the student has earned after t semesters.
We know that C(2) = 2y + x = 39 and C(5) = 5y + x = 75. We can use these two equations to solve for x and y:
2y + x = 39
5y + x = 75
Subtracting the first equation from the second equation gives:
3y = 36
Therefore, y = 12, which means the student takes 12 credits each semester. Substituting y = 12 into either of the two equations above gives x = 15, which means the student had 15 credits before starting college.
Thus, the equation that represents the number of credits the student has earned after t semesters is C(t) = 12t + 15.
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Help 100 pts. I don’t get it
The area of a flower bed is 24 square feet and the equation to represent the equal area is 24+20x+4x²=24.
Given that, the dimensions of rectangular flower bed are length= 6 feet and width = 4 feet.
Dimensions of large rectangle are Length = 6+2x and width = 4+2x
Now, area of large rectangle = (6+2x)×(4+2x)
= 24+8x+12x+4x²
Area of a rectangular flower bed = 6×4
= 24 square feet
Tulips and Daisy have same area to plant
So, the equation is 24+20x+4x²=24
4x²+20x=0
Therefore, the area of a flower bed is 24 square feet and the equation to represent the equal area is 24+20x+4x²=24.
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Given the vectors a and b. sketch the vector a+b
The resultant of the two vectors or the sum of the two vector is equal to the diagonal of parallelogram.
What is parallelogram law of vector addition?
Parallelogram law of vector addition states that if two vectors are represented in magnitude and direction by the adjacent sides of a parallelogram, the diagonal of the parallelogram drawn from the point of intersection of the vectors represents the resultant vector in magnitude and direction.
For the given vector a, and vector, the resultant of the two vector is calculated as;
-----------> a and --------------->b = a + b
The diagram represents the addition of the two vectors, a and b;
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8, 40, 200, ...
Find the 10th term.
Answer: 15,625,000
Step-by-step explanation: x5, 8, 40, 200, 1,000, 5,000, 25,000, 125,000, 625,000, 3,125,000, 15,625,000
PLEASE HELP (WILL GIVE BRAINLIEST)
Answer:d
Step-by-step explanation:
Three cakes are sliced into 20 pieces each. Each cake contains 1 gold ring What is the
probability one person who eats one piece of cake from each of the 3 cakes will find 3 gold
rings?
Answer:
1/60
Step-by-step explanation:
For each cake, there is a 1/20 chance to find the gold ring since it is in one of the 20 slices. The probability for each cake is independent of each other, so we multiply
1/20 x 1/20 x 1/20
=1/60
MA.912.G.6.6 Benchmark Pre-Test (Multiple Choice) answers
Where the information relating to the circle and its diameter are given, the equation of the circle would be: "(x 5)² + y2 = 81" (Option B)
Why is this so?The above is determined by the standard form equation of a circle which is:
(x- h) ² + (y - k)² = r²
Here,
(h, k) is the center of the circle and r is the radius
In this case, the center is (-5 , 0) and the diameter is 18, so the radius is half the diameter, whic is 18/ 2 = 9
So making the required substitutions, the expression becomes:
(x-(-5))² + (y-(0))² = 9²
Which becomes
(x 5)² + y2 = 81
Thus, it is correct to state that Option B is the equation of the circle.
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Full Question:
Although part of your question is missing, you might be referring to this full question:
The center of a circle is at (-5, 0), and the diameter of the circle is 18. Which of the following is the equation of the circle?
A. (x 5)² + y = 9
B. (x 5)² + y2 = 81
C. (x 5)² + y2 = 9
D. (x 5)² + y2 81
Rearrange the formula = 2√
3
and make ‘’ the subject.
Making x as a subject, we get equation as x=(y+2)/5.
The given equation is y=5x-2.
To make x as the subject
y+2=5x
x=(y+2)/5
Therefore, making x as a subject, we get equation as x=(y+2)/5.
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"Your question is incomplete, probably the complete question/missing part is:"
Rearrange the formula to make x the subject y=5x-2.
Last week, the value of an investment changed at a rate of -\$ 3.15−$3.15 each day. After how many days was the total charge in $ 12.60?−$12.60?
Based on the rate of change, the total charge will be $ 12.60 in 4 days
After how many days was the total charge in $ 12.60?From the question, we have the following parameters that can be used in our computation:
Rate of change of investment = $3.15
Value of the investment = $12.60
Using the above as a guide, we have the following:
Number of days = Value of the investment/Rate of change of investment
Substitute the known values in the above equation, so, we have the following representation
Number of days = 12.60/3.15
Evaluate
Number of days = 4
Hence, the number of days is 4
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Five sevenths of the fifty six images used as backgrounds on julia iPad were photos from dance class that she took herself. How many photos are of julia dance class? **Make sure you provide Your final answer as a fraction with a denominator of 56
The number of photos from Julia dance class = 5/7 or 40/56 photos.
We shall use mathematical operations to calculate the photos that are of Julia dance class
How do we use mathematical operations to calculate the photos?Five sevenths of the images are photos from dance class, meaning that 5/7 x 56 = 40 photos from dance class.
We express the answer as a fraction with a denominator of 56 like this:
Number of photos / Total number of photos = 40/56
Number of photos from dance class / Total number of photos = 5/7
Therefore, the fraction of photos that are of Julia dance class (as a fraction with a denominator of 56) = 5/7 or 40/56.
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FOUR CARS CRASH AT 5 MILES AN HOUR. Cost of the repairs are 422, 456, 401, and 215
COMPUTE RANGE
SAMPLE VARIANCE,
AND SAMPLE STANDARD DEVIATION
The range of the repairs to the cars is 241.
The sample variance is 11, 671.33
The sample standard deviation would be 108.03.
How to find the range ?Range = Maximum repair cost - Minimum repair cost
= 456 - 215
= 241
How to find the sample variance and standard deviation ?Find mean :
= (422 + 456 + 401 + 215) / 4
= 373. 5
Sample variance :
= Sum of ( each repair cost - mean ) ² / (Number of cars - 1)
= ( 2, 340.25 + 6, 812.25 + 756.25 + 25, 105.25 ) / ( 4 - 1 )
= 11, 671. 33
Sample standard deviation:
= √sample variance
= √11, 671. 33
= 108. 03
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The histograms display the frequency of temperatures in two different locations in a 30 day period. A histogram titled “Temps in Desert Landing” with When comparing the data, which measure of variability should be used for both sets of data to determine the location with the most consistent temperature?
The measure of variability that should be used for both sets of data to determine the location with the most consistent temperature is:
Range, because Flower Town is symmetric
How to Interpret the Histogram?A histogram is a visual representation of statistical data that makes use of rectangles to illustrate the frequency of data points over a range of successive numerical intervals of equal width. The independent variable and dependent variable are plotted along the horizontal axis and the vertical axis, respectively, in the most typical type of histogram.
Now, from the given histograms, we can see that Desert landing is skewed while Flower town is symmetrical entirely.
Thus, The measure of variability that should be used for both sets of data to determine the location with the most consistent temperature is:
Range, because Flower Town is symmetric
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