Answers
These [tex]N[/tex] outcomes make up the entire sample space, so
[tex]\displaystyle \sum_{k=1}^N P(e_k) = P(e_1) + P(e_2) + P(e_3) + \cdots + P(e_N) = 1[/tex]
We're given that [tex]P(e_{j+1}) = 2 P(e_j)[/tex] for all [tex]j\in\{1,2,\ldots,N-1\}[/tex], so
[tex]P(e_1) + 2 P(e_1) + 2^2 P(e_1) + \cdots + 2^{N-1} P(e_1) = 1 \\\\ \implies P(e_1) = \dfrac1{1 + 2 + 2^2 + \cdots + 2^{N-1}} = \dfrac1{2^N - 1}[/tex]
Then we can solve the recurrence relation to get the probability of the [tex]j[/tex]-th outcome,
[tex]P(e_{j+1}) = 2 P(e_j) = 2^2 P(e_{j-1}) = 2^3 P(e_{j-2}) = \cdots \\\\ \implies P(e_{j+1}) = 2^j P(e_1) \\\\ \implies P(e_j) = 2^{j-1} P(e_1) = \dfrac{2^{j-1}}{2^N - 1}[/tex]
The probability of getting this sequence of [tex]k[/tex] outcomes is then
[tex]\displaystyle P(E_k) = P(e_1) + P(e_2) + \cdots + P(e_k) = \sum_{j=1}^k \frac{2^{j-1}}{2^N-1} = \frac{2^k-1}{2^N-1}[/tex]
as required.
Some preliminary results: If [tex]S[/tex] is the sum of the first [tex]n[/tex] terms of a geometric series with first term [tex]a[/tex] and common ratio [tex]r[/tex], then
[tex]S = a + ar + ar^2 + \cdots + ar^{n-1}[/tex]
[tex]\implies rS = ar + ar^2 + ar^3 + \cdots + ar^n[/tex]
[tex]\implies S - rS = a(1 - r^n)[/tex]
[tex]\implies S = \dfrac{a(1 - r^n)}{1 - r}[/tex]
which gives us, for instance,
[tex]1 + 2 + 2^2 + \cdots + 2^{N-1} = \dfrac{1 - 2^N}{1 - 2} = 2^N-1[/tex]
Related Questions
What is the slope of the line shown below (3,-2) (2,4)?
A. -2
B. -1/2
C. 1/2
D. 2
Answers
Answer:
D. 2
Step-by-step explanation:
rise / run = slope
rise = 2
run = 1
2/1 = 2
slope = 2
Answer:
2 (Answer D)
Step-by-step explanation:
The green line has a positive slope, so we can immediately reject Answers A and B.
A line with a slope of +2 rises steeply as we move from left to right, whereas a line with slope +1/2 rises much less steeply. Also, it's quite clear, visually, that the slope of the given line is greater than 1. Thus, the slope of the given line is +2 (Answer D).
urgent question is below
Answers
i doooooooont knoooooooooooooooow iiiittttttttttttttttttttttt
You have loaned a friend %100 and will charge him 5% annual simple interest when he pay it back.
Answers
Answer:
bro nobody can figure this out you need to word the entire question with numbers and all. and proper English pls
Rob is a high school student taking both algebra and geometry. Last night he had 27 homework problems combined from the two classes, and he took a total of 90 minutes Which value of k would cause the system of linear equations 35 x + 14 y = 119 and 5 x + 2 y = k to have an infinite number of solutions?
3
7
17
21
Answers
A system of the equation to be Dependent Consistent System the system must have multiple solutions. The correct option is C.
What is a System of equation?Inconsistent System
A system of equations to have no real solution, the lines of the equations must be parallel to each other.
Consistent System
1. Dependent Consistent System
A system of the equation to be Dependent Consistent System the system must have multiple solutions for which the lines of the equation must be coinciding.
2. Independent Consistent System
A system of the equation to be Independent Consistent System the system must have one unique solution for which the lines of the equation must intersect at a particular.
For the system of equations to have infinite number of solutions, the lines of the equations must coincide, therefore, the line must overlap on each other.
For the lines to overlap the equations must be in a ratio. Therefore, the ratio of the left sides of the equations must be equal to the right side,
[tex]\dfrac{35x+14y}{5x+2y} = \dfrac{119}{k}\\\\\\\dfrac{7(5x+2y)}{5x+2y} = \dfrac{119}{k}\\\\\\\dfrac71 = \dfrac{119}{k}\\\\k = 17[/tex]
Hence, the value of k must be 17, therefore, the correct option is C.
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What is [3-8]-(12÷3+1)²
Answers
Answer:
-30
Step-by-step explanation:
What is [3-8]-(12÷3+1)²
[3-8]-(12÷3+1)² =
[3 - 8] - (4 + 1)² =
-5 - (5)² =
-5 - 25 =
-30
Triangle X Y Z is shown. The length of X Z is 12, the length of Y Z is 11, and the length of Y X is 6.
A scalene triangle has the lengths 6, 11, and 12. Keyla uses the law of cosines to find the measure of the largest angle. Complete her work and find the measure of angle Y to the nearest degree.
1. 122 = 112 + 62 − 2(11)(6)cos(Y)
2. 144 = 121 + 36 − (132)cos(Y)
3. 144 = 157 − (132)cos(Y)
4. −13 = −(132)cos(Y)
Answers
According to law of cosines if the lengths of triangle are 12,11,6 then the angle will be 114=157-132cos(Y)
Given the length of XZ is 12, length of YZ is 11 and the length of YX=6.
According to law of cosines the angle of a triangle when the length of sides are given can be written as:
[tex]c^{2} =a^{2} +b^{2} -2abcosd[/tex]
where c is the side opposite to the angle which we have to find and a and b are the rest sides.
We have to put the values of sides in the above law:
[tex]12^{2} =6^{2} +11^{2} -2*6*11cosY[/tex]
144=36+121-132 cos Y.
Hence the angle can be written as 144=36+121-132 cos Y.
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Determine the growth defined by the equation y=1.4(3.72)^x.
Answers
Answer:
the type of growth defined by the equation is Exponential growth.
Step-by-step explanation:
Exponential growth :-Exponential growth is a pattern of data that shows greater increases with passing time, creating the curve of an exponential function. It occurs when the instantaneous rate of change of a quantity with respect to time is proportional to the quantity itself
expression for exponential growth -
f(x) = k(1+t)^x
f(x) = exponential growth
k = initial amount
t = rate of growth
x = no of time interval
therefor,
the equation shown in the question represents exponential growth with increasing time.
hence, Exponential growth is the correct answer.
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find the x intercepts of the following linear function 3x-5y=-15
Answers
Answer:
(-5, 0)
Step-by-step explanation:
The y value of an x intercept is always 0. Therefore, we can substitute 0 for y in the equation to find the x intercept:
3x - 5(0) = -15
3x = -15
x = -5
The x intercept is (-5, 0)
For the inverse variation equation p = 8/v , what is the value of V when p = 1/2?
Answers
Substituting in p = 1/2,
[tex]\frac{1}{2}=\frac{8}{v}\\ \\ v=\boxed{16}[/tex]
A boat travels upstream for 42 miles in 3 hours and returns in 2 hours traveling downstream in a river. What is the rate of the boat in still water and the rate of the current?
The rate of the boat in still water is : miles per hour
The rate of the current is: miles per hour:
Answers
Answer:
17.5 mph
Step-by-step explanation:
Let the speed of the boat in still water be B mph
Let the rate of the current be R mph
When going upstream the effective speed is B-R mph. In terms of values given effective upstream speed = 42/3 = 14 mph
So,
B-R = 14 ... (1)
Similarly downstream speed is B + R (faster downstream)
B + R = 42/2 = 21 .... (2)
(1) + (2) ==> 2B = 35
B = 35/2 = 17.5 mph
Jimmy Page puts $30 into a savings account that promises 1% interest
compounded quarterly. Which formula would you use to determine the
account balance in 25 years?
Answers
In 25 years the account balance will be $38.51 if Jimmy Page puts $30 into a savings account that promises 1% interest compounded quarterly.
What is compound interest?It is defined as the interest on the principal value or deposit and the interest which is gained on the principal value in the previous year.
We can calculate the compound interest using the below formula:
[tex]\rm A = P(1+\dfrac{r}{n})^{nt}[/tex]
Where A = Final amount
P = Principal amount
r = annual rate of interest
n = how many times interest is compounded per year
t = How long the money is deposited or borrowed (in years)
P = $30
r = 1% = 0.01
n = 4
[tex]\rm A = 30(1+\dfrac{0.01}{4})^{4\times25}[/tex]
After calculating:
A = $38.51
Thus, in 25 years the account balance will be $38.51 if Jimmy Page puts $30 into a savings account that promises 1% interest compounded quarterly.
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[tex] \displaystyle \sum_{n = 1}^{ \infty } {4}^{ - n} [/tex]
evaluation of the sum above
Answers
Answer:
[tex]\dfrac{1}{3}[/tex]
Step-by-step explanation:
Given:
[tex]\displaystyle \sum^{\infty}_{n=1} 4^{-n}[/tex]
The sigma notation means to find the sum of the given geometric series where the first term is when n = 1 and the last term is when n = ∞.
To find the first term in the series, substitute n = 1 into the expression:
[tex]\implies a_1=4^{-1}=\dfrac{1}{4}[/tex]
The common ratio of the geometric series can be found by dividing one term by the previous term:
[tex]\implies r=\dfrac{a_2}{a_1}=\dfrac{4^{-2}}{4^{-1}}=\dfrac{1}{4}[/tex]
As | r | < 1 the series is convergent.
When a series is convergent, we can find its sum to infinity (the limit of the series).
Sum to infinity formula:
[tex]S_{\infty}=\dfrac{a}{1-r}[/tex]
where:
a is the first term in the seriesr is the common ratioSubstitute the found values of a and r into the formula:
[tex]S_{\infty}=\dfrac{\frac{1}{4}}{1-\frac{1}{4}}=\dfrac{1}{3}[/tex]
Therefore:
[tex]\displaystyle \sum^{\infty}_{n=1} 4^{-n}=\dfrac{1}{3}[/tex]
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THE LENGTH OF A PAINTING OF A TREE IS 2/9 OF THE LENGTH OF THE ACTUAL TREE. IF THE LENGTH OF THE PAINTING OF THE TREE IS 16 INCHES, WHAT IS THE LENGTH IN INCHES OF THE ACTUAL TREE
Answers
The length of the actual tree will be 72 inches.
Fraction is the portion of a total amount where the above part of the fraction is the denominator and the bottom part of the fraction is called the numerator.
The equation whose highest degree of the variable is 1 is called a linear equation.
Here given that the length of a painting of the tree is 2/9 of the actual length of the tree.
So mathematically the length of a painting= (2/9)*the length of actual tree
Let the length of the actual tree is x
then from above it is clear that the length of a painting of the tree= (2/9)x
Given the length of the painting is 16 inches.
So the linear equation will be (2/9)x= 16
⇒ x= 16/(2/9)
⇒x= 16*9/2
⇒x=72 inches
Therefore the length of the actual tree will be 72 inches.
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only 1st main ,please do solve this asap
Answers
Answer:
(i) 8b-32
(ii) 7z^3+12z^2-20z
(iii) –q+p
(iv) a+ab
(v) 8x^2y+8xy^2-4x^2-7y^2
(vi) 4y^2-3y
Step-by-step explanation:
Rewrite exponential expressions
This is from Khan academy I have to attach a PNG if you can help me solve it! Thank you!
Answers
Answer:
Step-by-step explanation:
please help with this question
Answers
If a line has a slope of 4 and goes through the point open parentheses short dash 1 comma 1 close parentheses, then the equation for the line in slope-intercept form is ______________.
a.)
y equals short dash 4 x minus 3
b.)
y equals 4 x plus 3
c.)
y equals 4 x plus 5
d.)
y equals short dash 4 x minus 5
Answers
You are given the slope = 4 and a point is (-1,1)
The equation of a line is written as y - mx + b where m is the slope and b is the y intercept
Using the slope you now have y = 4x + b
Solve for b by replacing x and y with the values of the point:
1 = 4(-1) + b
Simplify:
1 = -4 + b
Add 4 to both sides:
b = 5
The equation of the line is C.) y = 4x + 5
which is the correct graph for the inequality
Answers
What is the semiannual interest payments of a 30 year bond with a face value of $50 million and a stated annual interest rate of 4 percent?
Answers
Based on the stated annual interest rate and the face value of the bond, the semiannual payments will be $1,000,000.
How can the semiannual interest payment be found?The formula to find the semiannual payment is:
= (Face value x Stated annual interest rate) / 2 semi-annual periods per year
Solving gives:
= (50,000,000 x 4%) / 2
= 2,000,000 / 2
= $1,000,000
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Suppose the only measuring cup you own is a one-third measuring cup. Using this measuring cup how many scoops will you need to have 3 2/3 cups of sugar
Answers
Answer:
11
Step-by-step explanation:
for each full cup of sugar, you need 3 one-third measuring cups.
we have 3 full cups so 3×3=9
and another 2 for the 2/3, 9+2=11
so a total of 11 scoops
2. Ahmed has ¢15.25 and spent $7.50 at the
Total Shop. How much money does he have
left?
Answers
Answer:
D)¢ 8.75
Step-by-step explanation:
that is my answer please
18 / 48 hrs left
Divide and answer the question.
Answers
Answer: I don’t understand
Step-by-step explanation:
Cooling towers are used to remove or expel heat from a process. A cooling tower's walls are modeled by x squared over 324 minus quantity y minus 90 end quantity squared over 1600 equals 1 comma where the measurements are in meters. What is the width of the cooling tower at the base of the structure? Round your answer to the nearest whole number.
36 meters
48 meters
80 meters
89 meters
Answers
The width of the cooling tower at the base of the structure will be A. 36 meters.
How to calculate the width?From the information given, the towers walls are modeled by x²/324 - (y² - 90)²/1600.
Therefore, the width of the cooling tower at the base of the structure will be:
= 2 × ✓324
= 2 × 18
= 36
In conclusion, the width is 36 meters.
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Answer:
It is actually D
Step-by-step explanation:
Eugene and Shanice each improved their yards by planting sod and geraniums. They bought their supplies from the
same store. Eugene spent $36 on 9 ft² of sod (s) and 2 geraniums (g). Shanice spent $63 on 9 ft² of sod and 5
geraniums. What is the solution to the system of equations? What is the cost of one ft² of sod and the cost of one
geranium?
Answers
The cost of one ft² of sod is $2 and the cost of one geranium is $9.
What are the linear equations that represent the question?9s + 2g = 36 equation 1
9s + 5g = 63 equation 2
What is the cost of one feet of sod and geranium?
Subtract equation 1 from equation 2:
3g = 27
g = 27 / 3
g = $9
Substitute for g in equation 1
9s + (2 x 9) = 36
9s + 18 = 36
9s = 36 - 18
9s = 18
s = 18 / 9
s = $2
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For the function f(x)=|x−7|−4, evaluate f(x+1).
Answers
Answer:
[tex]\displaystyle{f(x+1)=|x-6|-4}[/tex]
Step-by-step explanation:
To evaluate [tex]\displaystyle{f(x+1)}[/tex], substitute [tex]\displaystyle{x}[/tex] to [tex]\displaystyle{x+1}[/tex] so we should have:
[tex]\displaystyle{f(x+1) = |x+1-7|-4}[/tex]
Evaluate the expression and finally, we have:
[tex]\displaystyle{f(x+1)=|x-6|-4}[/tex]
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https://brainly.com/question/27923727 - How to find inverse and value of function
pls see the attachment
Answers
24 of the 720 numbers have the numbers 1, 2 and 3 in that order
How to determine the number of arrangements?The digits are given as:
1, 2, 3, 4, 5, and 6
For 1, 2 and 3 to come in that order, then the three numbers must be treated as a digit
So, we have:
(123), 4, 5, and 6
Now, there are 4 digits.
The number of arrangements is then calculated as:
Arrangements = 4!
Expand
Arrangements = 4 * 3 * 2 * 1
Evaluate
Arrangements = 24
Hence, 24 of the 720 numbers have the numbers 1, 2 and 3 in that order
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1. (y + x) = 4; use x = 2 and y = 2
2. x + y = 6; use x = 5 and y = 6
3. y(x + 5); use x = 1 and y = 4
4. q-m+q; use m = 3 and q = 6
Answers
4(1+5)= 24
6-3+6= 9 or 3
What is the surface area of prism below?
Answers
Answer:
Aprism = 483.246 yd2
Step-by-step explanation:
The surface area is the sum of the 2 circles areas and the rectangle area.
Ac1 = PI*r2
Ac1 = (3.14)*(11.4/2)2
Ac1 = 102.0186 yd2
Ac1 = Ac2 then
Ac2 = 102.0186 yd2
Now the perimeter of the circle is one the sides of the rectangle:
Pc = 2PI*r
Pc = 2*3.14*(11.4/2)
Pc = 35.796 yd
Now the area of the rectangle is:
Ar = 7.8yd*35.796yd
Ar = 279.2088 yd2
Then, the surface area of prism is:
Aprism = Ac1 + Ac2 + ArAprism = 102.0186yd2 + 102.0186yd2 + 279.2088yd2Aprism = 483.246 yd2Complete the following statement of congruence:
A. TRS
B. RTS
C. STR
D. RST
Answers
Answer:
B. △RTS
Step-by-step explanation:
△MON ≅ △RTS
y = 4/5 + square root x + 0.2 - 5x
Answers
Answer:
5
Step-by-step explanation:
