suppose that, as in exercises 5.11 and 5.79, y1 and y2 are uniformly distributed over the triangle shaded in the accompanying diagram. (–1, 0) (1, 0) (0, 1) y1 y2 a find cov(y1, y2). b are y1 and y2 independent? (see exercise 5.55.) c find the coefficient of correlation for y1 and y2. d does your answer to part (b) lead you to doubt your answer to part (a)? why or why not?
Answers
Using the definition of covariance, Cov(Y(1)Y(2))= 0. From the given information the two variables Y(1), and Y(2), are dependent and, p(y(1)y(2)) ≠ P(y(1))p(y(2)). So, it can be concluded that Uncorrelated variables need not be independent.
Given joint probability function of Y(1) and Y(2), is
p(Y(1)*Y(2)) = 1/3, for (y(1)*y(2)) = (- 1, 0), (0, 1), (1, 0)
So Y(1), takes random variable -1,0,1.
And Y(2) takes random variable 0,1.
Y(2)|Y(1) -1 0 1
0 1/3 0 1/3
1 0 1/3 0
From the table:
The marginal probability function of Y(1) is;
P(Y(1)=y(1)) = ∑P(Y(1)=y(1)), y(1)= -1,0,1. That is;
Y(1)=y(1) -1 0 1
P(Y(1)=y(1)) 1/3 1/3 1/3
From the definition of expectation,
E[y(1)]= [tex]\Sigma_{y_{1}}[/tex] y(1)P(Y(1)=y(1))
E[y(1)]= -1(1/3)+0(1/3)+1(1/3)
E[y(1)]= -1/3+0+1/3
E[y(1)]= 0
The marginal probability mass function of y(2) is
P(Y(2)=y(2)) = ∑P(y(1)y(2)), y(2)= 0,1. That is;
Y(2)=y(2) 0 1
[tex]P_{Y(2)}[/tex](y(2)) 2/3 1/3
Now E[y(2)]= [tex]\Sigma_{y_{2}}[/tex] y(2)P(Y(2)=y(2))
E[y(2)]= 0(2/3)+1(1/3)
E[y(2)]= 1/3
And
E[y(1)y(2)]= [tex]\Sigma_{y_{1}}\Sigma_{y_{2}}[/tex] y(1)y(2)P(y(1)y(2))
E[y(1)y(2)]= -1(0)(1/3)+0(1)(1/3)+1(0)(1/3)
E[y(1)y(2)]= 0
From the definition of covariance.
The Cov(Y(1)Y(2))= E[Y(1)Y(2)]-E[Y(1)E[Y(2)]
Cov(Y(1)Y(2))= 0-1/3 (0)
Cov(Y(1)Y(2))= 0
The Cov(Y(1)Y(2))=0, implies that the two variables are uncorrelated. But from the given information the two variables Y(1), and Y(2), are dependent and, p(y(1)y(2)) ≠ P(y(1))p(y(2))
Hence, it can be concluded that Uncorrelated variables need not be independent.
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Related Questions
What is the average rate of change of the function f(x)=2(3)xfrom x = 2 to x = 4?.
Answers
The average rate of change of the function f(x)=2(3)x from x = 2 to x = 4 = 72
How do functions work?The term "function" refers to the relationship between a collection of inputs and outputs with one each. Simply described, a function is an association of inputs where each input is coupled with exactly one output. Every function has an associated domain, codomain, or range. The most common way to represent a function is as f(x), where x represents the input.
To calculate the average rate of change, use the slope formula:
y2−y1/x2−x1=m
f(4)−f(2)/4−2=m
Finding the values of f(4) and f(2) will help us to simplify this.
f(4)=2(3)4=2(81)=162
f(2)=2(3)2=2(9)=18
m=162−18/2
m=144/2
m=72
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a binomial distribution has a 60% rate of success. there are 18 trials. what is the probablility that there will be at least 12 successes
Answers
The probability that there will be at least 12 successes in the total 18 trials with 60% rate of success is ¹⁸C₁₂(0.6)¹²(0.4)⁶+¹⁸C₁₃(0.6)¹³(0.4)⁵+¹⁸C₁₄(0.6)¹⁴(0.4)⁴+ ¹⁸C₁₅(0.6)¹⁵(0.4)³ +¹⁸C₁₆(0.6)¹⁶(0.4)²+ ¹⁸C₁₇(0.6)¹⁷(0.4)¹+ ¹⁸C₁₈(0.6)¹⁸(0.4)⁰
Binomial Probability distribution, In probability theory and statistics, the binomial distribution with parameters n and p is the discrete probability distribution of the number of successes in a series of n independent experiments.
We have given that :
Rate of sucess (p) = 60% = 0.6
Rate of failure (q) = 1 - 0.6 = 0.4
Number of trials (n) = 18
We want to find P(there will be at least 12 sucess)
Use binomial distribution to find given mentioned probability.
X ~ Bin( 18, 0.6)
P (X= x ) = ⁿCₓ(p)ˣ(q)⁽ⁿ⁻ˣ⁾
Let, X= Number of success among 18
So that it suitable for binomial distribution with parameters are n and p.
P(there will be atleast 12 success)
= P(X>= 12)
18
= ∑ ⁿCₓ (p)ˣ(q)⁽ⁿ⁻ˣ⁾ =
x = 12
¹⁸C₁₂(0.6)¹²(0.4)⁶+¹⁸C₁₃(0.6)¹³(0.4)⁵+¹⁸C₁₄(0.6)¹⁴(0.4)⁴+ ¹⁸C₁₅(0.6)¹⁵(0.4)³ +¹⁸C₁₆(0.6)¹⁶(0.4)²+ ¹⁸C₁₇(0.6)¹⁷(0.4)¹+ ¹⁸C₁₈(0.6)¹⁸(0.4)⁰
= say x
Hence, the required probability is x
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the difference of vidya's age and 15 is 66 use the variable v to represent vidya's age
Answers
Answer:
[tex]\huge\boxed{\sf v = 81}[/tex]
Step-by-step explanation:
Let Vidya's age be v.
Given condition:v - 15 = 66
Add 15 to both sides
v = 66 + 15
v = 81[tex]\rule[225]{225}{2}[/tex]
Given AB, find C which is 1/3 of the distance from A to B. A(1,2) B(-5,8)
Answers
Answer: (2,0)
Step-by-step explanation:
Danni placed $5700 in a savings account which compounds interest continuously at a rate of 2. 1%. How much will she have in the account after 4 years? round your answer to the nearest whole number. Do not include a $ in your answer.
Answers
Using the compound interest method, we can calculate that Danni will have $6,194.09 in her account after 4 years.
What is compound interest?Compound interest, also known as interest on principal and interest, is the practice of adding interest to the principal amount of a loan or deposit.
Compound interest works by periodically adding collected interest to your main, which is the sum you first deposited into the savings account so that interest can then be earned on both.
In essence, your interest begins to generate interest of its own.
Compound interest formula: [tex]A=P\left(1+\frac{r}{100}\right)^{n t}[/tex]
Now, calculate the amount Danni will have after 4 years:
[tex]A=P\left(1+\frac{r}{100}\right)^{n t}[/tex]
A = 5700(1+0.021)⁴
A = 5700(1.0210⁴
A = $6,194.09
Therefore, using the compound interest method, we can calculate that Danni will have $6,194.09 in her account after 4 years.
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Answer:
A=6199.48
Step-by-step explanation:
Use the formula for calculating compound interest A=P0ert where A is the unknown, P0=5700, r=0.021, and t=4. Substitute the values into the formula and simplify.
A=5700e0.021⋅4
A=5700e0.084
A=5700(1.0876...)
A=6199.48
The amount in the account after four years is A≈6199, to the nearest dollar.
Olivia has has read 40 pages of a 70 page book, 60 pages of an 85 page book, and 43 pages. What is the percentage f pages that Olivia has not read?
Answers
Answer:
Below
Step-by-step explanation:
Total pages read = 40 + 60 +43 = 143 pages read
Total pages of all of the books = 70 + 85 + x
percentage READ = (143) / ( 70 + 85 + x) * 100% = ???
percentage NOT read = 100 - ???
(since 'x' was not given in the question post)
Read pages:
40+60+43=143
All pages:
70+85+43=198
198-143=55
55/198=0,2778
0,2778×100%=27.78%
Find the missing value. Round your answer to the nearest tenth of a percent. Principal=$1300, Rate=?, Time=18 months, Simple Interest=$90
Answers
The missing value in the simple interests formula is rate and it is solved to be 4.62%
How to find the rateThe formula for simple interest is
= P * T * R / 100
P = principal= $1300
T = time = 18 months = 1.5 years
R = rate = ?
simple interest = $90
substituting the values
90 = 1300 * 1.5 * R / 100
90 * 100 = 1300 * 1.5 * R
9000 = 1950R
R = 9000 / 1950
R = 4.615385
R = 4.62
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I would like to create a rectangular vegetable patch. The fencing for the east and west sides costs $4 per foot, and the fencing for the north and south sides costs only $2 per foot. I have a budget of $96 for the project. What are the dimensions of the vegetable patch with the largest area i can enclose?.
Answers
The dimensions of the vegetable patch with the largest area we can enclose are
length = 12 feet and breadth = 6 feet.
Given, a rectangular vegetable patch.
The fencing for the east and west sides costs $4 per foot, and the fencing for the north and south sides costs only $2 per foot.
We have a budget of $96 for the project.
Let the length of the rectangle be, l
and breadth of the rectangle be, b
according to the ques,
4l + 8b = 96
l + 2b = 24
l = 24 - 2b
Now, area of the rectangle be,
Area = length×breadth
Area = l×b
A = (24 - 2b)b
A = 24b - 2b²
On differentiating both sides, we get
A' = 24 - 4b
b = 6
l = 24 - 12 = 12
So, the dimensions of the vegetable patch with the largest area we can enclose are
length = 12 feet and breadth = 6 feet.
Hence, The dimensions of the vegetable patch with the largest area we can enclose are
length = 12 feet and breadth = 6 feet.
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Simplify: 8x62x4
A.6x2
B.4x2
C.8x10
D.10x10
Answers
in regression analysis, measures the proportion of the variation in the dependent variable that is explained by the independent variable(s).
Answers
The coefficient of determination (R² or r-squared) is a statistical measure in a regression model that determines the proportion of variance in the dependent variable that can be explained by the independent variable.
What is variance?Variance is a measure of dispersion that, in contrast to range and interquartile range, accounts for the spread of all data points in a data collection. Along with the standard deviation, which is just the square root of the variance, it is the measure of dispersion that is most frequently employed. The statistical assessment of the variation in numbers within a data collection is known as variance. In more detail, variance assesses how far apart each number in the collection is from the mean (average) and, consequently, from each other.
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Which is the best estimate of the
solution for the system of linear
equations graphed below?
A.(0.5,2)
B. (1.5, 2)
C. (2,1.5)
D. (2.5, 1.5)
Answers
Answer:
B. (1.5, 2)
Step-by-step explanation:
Hope it helps!
Answer:
B
Step-by-step explanation:
The intersection of the two lines looks to be about (1.5 , 2)
question: 14. which of the following shapes replaces the question mark to complete the series? ? 8 oo aa
Answers
Pattern Recognition is defined as the pattern is a number, shape, or something repeated in a particular way. Analysis of patterns should consider on different basis. The correct option is option (d).
The main basis of Pattern Recognition are the following:
1. Color change
2. Shape change
3. Rotation change
4. Size change
we have given three shapes pattern and we have to find out the perfect shape for forth place and the required shape follow the order like others.
After examine the pattern, we can see that the first figure is a filled triangle and the second figure is a hollow circle. This pattern is maintained alternately. The last figure(3rd place) is an open circle, so the next figure is a filled triangle.
So, the correct option is d.
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Please Help what is x
Answers
Answer: x=[tex]8\sqrt{2}[/tex]
Step-by-step explanation:
There are special right triangles in this! First, you need to find out what I am saying.
45-45-90 triangle
The bottom triangle has angles measures of 45, 45, and 90 degrees. There is a theorem that states that the two legs are equal, and the hypotenuse is leg×[tex]\sqrt{2}[/tex].
Plugging it in will give you leg lengths of [tex]4\sqrt{2}[/tex].
Plug in that number into the side that connects with our next very special triangle:
30-60-90 triangle
Another theorem states that the hypotenuse is shorter leg times 2, the longer leg is shorter leg times [tex]\sqrt{3}[/tex].
Great! We have the shorter leg. Now we need to find the hypotenuse. Multiply [tex]4\sqrt{2}[/tex] by 2, and you get [tex]8\sqrt{2}[/tex]. That is x!
The final answer to this problem is: x=[tex]8\sqrt{2}[/tex]
Hope this helped a lot!
3x + 4x +5x = 4(x + 2) Please write as the actual answer and a decimal
Answers
Answer: 1
Step-by-step explanation:
We can rewrite the equation as
12x = 4x +8
8x= 8
x=1
Helppppppppppppppppppppppppp
Answers
Then multiply by the negative for -16/49
Answer:
-[tex]\frac{16}{49}[/tex]
Step-by-step explanation:
First do: (-[tex]\frac{4}{7}) ^{2}[/tex], which is -[tex]\frac{4}{7}[/tex]*-[tex]\frac{4}{7}[/tex] which is [tex]\frac{16}{49}[/tex] and then add in the other negative: -[tex]\frac{16}{49}[/tex]
In two or more complete sentences, describe how to determine if the function f(x)=2cos(x) is even by looking at a graph.
Answers
The given function f(x) = 2 cos (x) is an even function because the result of the cosine of negative angles are the same with those of positive angles.
How to determine an Even Function?
To check if a function is odd or even, we have to calculate f(−x).
The function is even if f(-x) = f(x)
The function is odd if f(-x) = -f(x)
We are given the function as; f(x) = 2 cos (x)
f(-x) = 2 cos (-x)
Now, we know that the cosine of a negative angle always gives the same value as the cosine of the positive angle and as such, we can say that;
2 cos (-x) = 2cos (x)
The same result as f(x) which tells us that it is an even function
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Francisco is building a window seat below a bay window in the shape of an isosceles trapezoid. His blueprint for the window seat is shown below.
What is the area of the window seat?
Answers
Answer:
56
Step-by-step explanation:
Five bands in a tour agree to determine the order of performance based on random selection. The names are placed on cards and the cards are pulled randomly.
What is the probability of Ultraviolet performing fourth and Abacus performing last?
A: 3/5
B: 7/120
C: 7/32
D: 1/20
Answers
The probability of Ultraviolet performing fourth and Abacus performing last is 1/20.
What is probability ?Probability shows possibility to happen an event, it defines that an event will occur or not. The probability varies from 0 to 1.
Given that,
In a tour, five bands agree to determine the order of performance on the basis of random selection.
Since, the name of 5 bands are pulled randomly.
The probability of Ultraviolet performing fourth = 1/5
And the probability of Abacus performing last = 1/4, since 1 place is fixed for Ultraviolet.
Probability of Ultraviolet performing fourth and Abacus performing last
= 1/5 x 1/4 = 1/20
The required probability is 1/20.
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Can someone please help me
Answers
Answer:
6
Step-by-step explanation:
Perimeter = 2l + 2w
48 = 2(18) + 2(w)
48 = 36 + 2w
12 = 2w
w = 6
Could you please give me brainliest for this? Thank you!
use the product rule to find the derivative of the given function. b. find the derivative by expanding the product first.
Answers
The derivative of the given function using product rule is 6x+1 .
What is Product Rule?
This rule is used when we want to differentiate a function that may be regarded as the product of one or more functions.
Let u(x) and v(x) be differentiable functions.Then the product of the functions u(x)v(x) is also differentiable and: (uv)′=u′v+uv′
Main Body:
We add the derivative of the first times the second to the first times the
derivative of the second.
Power Rule:
If f(x)=xn
where n is any real number, then:
(xⁿ)′=nxⁿ⁻¹
. We have,
f(x) = (x-1) (3x+4)
Differentiate with respect to z by using product rule
(uv)′=u′v+uv′
h' (x) = (x-1)' (3x+4) +(x-1)(3x+4)'
h'(X) = 1(3x+4) + (x-1) (3)
On simplifying:
h'(x) = 3x+4 +3x-3
h'(x) = 6x +1
Hence the derivative of function is 6x+1.
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Solve 3x² - x = 10
A.X=2 and X=-15
B. X=-5 and X= 2/3
C. X=-5/3 and X=2
D. X=-3 and X=10
Answers
Answer:
C. x = -5/3 and x = 2
Step-by-step explanation:
Bring the 10 over to the other side of the equation.
3x² - x - 10 = 0
Factorise.
(3x + 5)(x - 2) = 0
3x + 5 = 0 or x - 2 = 0
3x = -5 or x = 2
x = -5/3 or x = 2
If x4xy= x16, what is the value of y?
Answers
Answer:
4/x
Step-by-step explanation:
a tower that is 106 feet tall casts a shadow 135 feet long. find the angle of elevation of the sun to the nearest degree.
Answers
The angle of elevation of the sun is [tex]tan^{-1}[/tex](106/ 135).
The angle of elevation:
The angle is formed by the line of sight and the horizontal plane for an object above the horizontal.
Here we have to find the angle of elevation of the sun.
Data given:
length of the tower = 106 feet
shadow form by the tower = 135 feet
The formula for the angle of elevation = height/base
tanФ = height/base
= 106/ 135
Ф = [tex]tan^{-1}[/tex](106/135)
Therefore the angle of elevation is [tex]tan^{-1}[/tex]( 106/ 135).
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Two towns have accumulated different amount of snow. IN town 1, the snow depth is increasing by 3 1/2 inches every hour, and currently has 5 inches of snow. IN town 2, the snow depth is increasing by 2 1/4 inches every hour, and currently has 6 inches of snow. In how many hours will the snowfall of both towns by equal.
Answers
The snowfall of both towns by equal in 4/5 hours or 48 minutes.
What is a linear equation?A linear equation is an algebraic equation of degree one. In general, the variable or the variables(in the case of a linear equation in two variables) the variables are x and y.
Assuming the no. of hours to be x, therefore the expression for town 1 is
y = (7/2)x + 5 and the expression for town 2 is y = (9/4)x + 6.
The snowfall of both towns by equal when,
(7/2)x + 5 = (9/4)x + 6.
(7/2)x - (9/4)x = 6 - 5.
[(14 - 9)x]/4 = 1.
(5/4)x = 1.
x = 4/5 hours Or (4/5)×60 = 48 minutes the snowfall of both towns by equal.
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Find the area of the composite figure. I got 78 but my teacher said that it is incorrect. WILL MARK BRAINLIEST! Thanks!
Answers
Answer:
44
Step-by-step explanation:
Essentially there are 3 triangles
The top left has a base of 5 and a height of 4
Its area is 1/2 x b x h = 1/2 x 5 x 4 = 10
Then there are two triangles with the same base of 2.5 + 6 = 8.5 and same height of 4 feet. So each of the triangles has the same area as the other.
Total area of both triangles = 2 (1/2 x b x h) = bh = 8.5 x 4 = 34
Total area = 10 + 34 = 44
Answer:
44 ft²
Step-by-step explanation:
Here's how I divided the parts of the composite figure:
1. There is a triangle at the left with the vertex pointing down and base above. The base is 5 ft. The height is 4 ft.
2. There are 4 triangles left. I divided them into two triangles like this:
An upper triangle which has a base of 2.5 ft + 6 ft, and a height of 4 ft with a horizontal base on the bottom and a vertex on the upper side.
A lower triangle that is a mirror image of the triangle just above. The dimensions are the same.
Triangle on upper left:
base = 5 ft
height = 4 ft
Triangles at right (2 triangles with the same dimensions)
base = 2.5 ft + 6 ft = 8.5 ft
height = 4 ft
area of triangle = base × height / 2
Total Area = (5 ft)(4 ft)/2 + 2 × (8.5 ft)(4 ft)/2
Total Area = 10 ft² + 34 ft²
Total Area = 44 ft²
Triangle PQR has the following measures: p= 16.2 m, q = 35.0 m, and m/R = 75°.
What is the area of triangle PQR?
Answers
After solving the area of the triangle PQR obtained will be equal to 273.6 m²
What is a triangle?
The basic polygonal shape of a triangle has three sides & three interior angles. This is one of the fundamental shapes in geometry and is particularly associated. It has three connected vertices. Triangles can be divided into a variety of categories according to their sides and angles.
As per the given information given in the question,
p= 16.2
q = 35.0 m and,
m ∠R = 75°
Find x by cosine rule,
cos 75° = (16.2² + 35² - x²)/(2 × 16.5 × 35)
x = 34.5 m
The value of r is 34.5 mm then the third side of the triangle pqr will be,
Use heron's formula,
s = (16.2 + 35 + 34.5)/2
s = 42.85
Area = [tex]\sqrt{s(s-p)(s-q)(s-r)}[/tex]
A = √42.85(42.85 - 16.2)(42.85 - 35)(42.85 - 34.5)
A = 273.6 m²
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the base of the first booth is the shape of a triangle. write a polynomial that represents the area of the booth. write your answer in descending order. please use the palette below to enter your answer.
Answers
The polynomial that represents the area of the booth is 3[tex]t^3[/tex]-12t.
How to calculate polynomial for triangle area?Polynomial is a expression of variable, coefficients, and constants with mathematics operation of subtraction, addition, multiplication, and powers (only positive integers) of variable.
So, basically to calculate polynomials is the same as normal calculations except that the numbers are changed to variables.
Base length = 2[tex]t^2[/tex]-8
Height length = 3t
Calculate the area of triangle. So,
Area of triangle = (1/2) × base × height
= (1/2) × 2[tex]t^2[/tex]-8 × 3t
= (1/2) × 6[tex]t^3[/tex]-24t
= 3[tex]t^3[/tex]-12t
Thus, the area of the booth can be written by this polynomial 3[tex]t^3[/tex]-12t.
Your question is incomplete, but most probably your full question was (image attached).
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Someone answer this please thank you
Answers
Answer:
The answer is 1/2
Step-by-step explanation:
x1 y1. x2. y2
(5, 8) (2, 2)
So you suppose to subtract xl by x2 then subtract y1 by y2
5-2.
------=
8-2.
which equal 3/6. Which 3/6 is equal to 1/2
33, 25, 42, 25, 31, 37, 46, 29, 38 what is the interquartile range of the data?
Answers
The interquartile range of the data will be 13
When arranged from lowest to highest, the IQR reflects the median 50% of values. To calculate the interquartile range (IQR), firstly compute the median (middle value) of the data's lower and upper halves. All those are quartile 1 (Q1) and quartile 3 (Q3) values (Q3). The interquartile range is the difference between quarters three and then one.
first, let us sort the data from lowest to highest
25,25,29,31,33,37,38,42,46
Q1- median of the lower half of the data=(29+25)/2=27
Q3-median of the upper half of the data=(38+42)/2=40
the interquartile range of the data will be Q3-Q1
40-27= 13
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9 2 8 7 6 5 1 4 3 show the contents of the array above each time an insertion sort changes it while sorting the array into ascending order.
Answers
The array that has been sorted using insertion sorting 1 2 3 4 5 6 7 8 9.
A straightforward sorting algorithm called insertion sort functions like how you might arrange playing cards in your hands.
Initial Pass: 2 9 8 7 6 5 1 4 3
The array's first two members are initially compared using an insertion sort.
Since 9 is bigger than 2, they are not arranged in ascending order, and 2 is not in the proper place. So, switch 9 and 2.
Therefore, 2 is currently kept in a sorted sub-array.
Second Pass: Continue with the next two elements and compare them.
2 8 9 7 6 5 1 4 3
Here, 9 is bigger than 8, therefore both items appear to be in ascending order; as a result, there won't be any switching. Along with 8, 9 is also kept in a sorted sub-array.
Third pass: Currently, the sorted sub-array has two entries, which are 2 and 8.
Moving on to the following two components, which are 9 & 7,
2 7 8 9 6 5 1 4 3
Continue sorting similarly; the results would be 1 2 3 4 5 6 7 8 9.
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