Annual windstorm losses, X and Y, in two different regions are independent, and each is uniformly distributed on the interval [0, 10]. Calculate the covariance of X and Y, given that X+ Y < 10.
Answers
Answer:
[tex]Cov(X,Y) = -\frac{ 25}{9}[/tex]
Step-by-step explanation:
Given
[tex]Interval =[0,10][/tex]
[tex]X + Y < 10[/tex]
Required
[tex]Cov(X,Y)[/tex]
First, we calculate the joint distribution of X and Y
Plot [tex]X + Y < 10[/tex]
So, the joint pdf is:
[tex]f(X,Y) = \frac{1}{Area}[/tex] --- i.e. the area of the shaded region
The shaded area is a triangle that has: height = 10; width = 10
So, we have:
[tex]f(X,Y) = \frac{1}{0.5 * 10 * 10}[/tex]
[tex]f(X,Y) = \frac{1}{50}[/tex]
[tex]Cov(X,Y)[/tex] is calculated as:
[tex]Cov(X,Y) = E(XY) - E(X) \cdot E(Y)[/tex]
Calculate E(XY)
[tex]E(XY) =\int\limits^X_0 {\int\limits^Y_0 {\frac{XY}{50}} \, dY} \, dX[/tex]
[tex]X + Y < 10[/tex]
Make Y the subject
[tex]Y < 10 - X[/tex]
So, we have:
[tex]E(XY) =\int\limits^{10}_0 {\int\limits^{10 - X}_0 {\frac{XY}{50}} \, dY} \, dX[/tex]
Rewrite as:
[tex]E(XY) =\frac{1}{50}\int\limits^{10}_0 {\int\limits^{10 - X}_0 {XY}} \, dY} \, dX[/tex]
Integrate Y
[tex]E(XY) =\frac{1}{50}\int\limits^{10}_0 {\frac{XY^2}{2}}} }|\limits^{10 - X}_0 \, dX[/tex]
Expand
[tex]E(XY) =\frac{1}{50}\int\limits^{10}_0 {\frac{X(10 - X)^2}{2} - \frac{X(0)^2}{2}}} }\ dX[/tex]
[tex]E(XY) =\frac{1}{50}\int\limits^{10}_0 {\frac{X(10 - X)^2}{2}}} }\ dX[/tex]
Rewrite as:
[tex]E(XY) =\frac{1}{100}\int\limits^{10}_0 X(10 - X)^2\ dX[/tex]
Expand
[tex]E(XY) =\frac{1}{100}\int\limits^{10}_0 X*(100 - 20X + X^2)\ dX[/tex]
[tex]E(XY) =\frac{1}{100}\int\limits^{10}_0 100X - 20X^2 + X^3\ dX[/tex]
Integrate
[tex]E(XY) =\frac{1}{100} [\frac{100X^2}{2} - \frac{20X^3}{3} + \frac{X^4}{4}]|\limits^{10}_0[/tex]
Expand
[tex]E(XY) =\frac{1}{100} ([\frac{100*10^2}{2} - \frac{20*10^3}{3} + \frac{10^4}{4}] - [\frac{100*0^2}{2} - \frac{20*0^3}{3} + \frac{0^4}{4}])[/tex]
[tex]E(XY) =\frac{1}{100} ([\frac{10000}{2} - \frac{20000}{3} + \frac{10000}{4}] - 0)[/tex]
[tex]E(XY) =\frac{1}{100} ([5000 - \frac{20000}{3} + 2500])[/tex]
[tex]E(XY) =50 - \frac{200}{3} + 25[/tex]
Take LCM
[tex]E(XY) = \frac{150-200+75}{3}[/tex]
[tex]E(XY) = \frac{25}{3}[/tex]
Calculate E(X)
[tex]E(X) =\int\limits^{10}_0 {\int\limits^{10 - X}_0 {\frac{X}{50}}} \, dY} \, dX[/tex]
Rewrite as:
[tex]E(X) =\frac{1}{50}\int\limits^{10}_0 {\int\limits^{10 - X}_0 {X}} \, dY} \, dX[/tex]
Integrate Y
[tex]E(X) =\frac{1}{50}\int\limits^{10}_0 { (X*Y)|\limits^{10 - X}_0 \, dX[/tex]
Expand
[tex]E(X) =\frac{1}{50}\int\limits^{10}_0 ( [X*(10 - X)] - [X * 0])\ dX[/tex]
[tex]E(X) =\frac{1}{50}\int\limits^{10}_0 ( [X*(10 - X)]\ dX[/tex]
[tex]E(X) =\frac{1}{50}\int\limits^{10}_0 10X - X^2\ dX[/tex]
Integrate
[tex]E(X) =\frac{1}{50}(5X^2 - \frac{1}{3}X^3)|\limits^{10}_0[/tex]
Expand
[tex]E(X) =\frac{1}{50}[(5*10^2 - \frac{1}{3}*10^3)-(5*0^2 - \frac{1}{3}*0^3)][/tex]
[tex]E(X) =\frac{1}{50}[5*100 - \frac{1}{3}*10^3][/tex]
[tex]E(X) =\frac{1}{50}[500 - \frac{1000}{3}][/tex]
[tex]E(X) = 10- \frac{20}{3}[/tex]
Take LCM
[tex]E(X) = \frac{30-20}{3}[/tex]
[tex]E(X) = \frac{10}{3}[/tex]
Calculate E(Y)
[tex]E(Y) =\int\limits^{10}_0 {\int\limits^{10 - X}_0 {\frac{Y}{50}}} \, dY} \, dX[/tex]
Rewrite as:
[tex]E(Y) =\frac{1}{50}\int\limits^{10}_0 {\int\limits^{10 - X}_0 {Y}} \, dY} \, dX[/tex]
Integrate Y
[tex]E(Y) =\frac{1}{50}\int\limits^{10}_0 { (\frac{Y^2}{2})|\limits^{10 - X}_0 \, dX[/tex]
Expand
[tex]E(Y) =\frac{1}{50}\int\limits^{10}_0 ( [\frac{(10 - X)^2}{2}] - [\frac{(0)^2}{2}])\ dX[/tex]
[tex]E(Y) =\frac{1}{50}\int\limits^{10}_0 ( [\frac{(10 - X)^2}{2}] )\ dX[/tex]
[tex]E(Y) =\frac{1}{50}\int\limits^{10}_0 [\frac{100 - 20X + X^2}{2}] \ dX[/tex]
Rewrite as:
[tex]E(Y) =\frac{1}{100}\int\limits^{10}_0 [100 - 20X + X^2] \ dX[/tex]
Integrate
[tex]E(Y) =\frac{1}{100}( [100X - 10X^2 + \frac{1}{3}X^3]|\limits^{10}_0)[/tex]
Expand
[tex]E(Y) =\frac{1}{100}( [100*10 - 10*10^2 + \frac{1}{3}*10^3] -[100*0 - 10*0^2 + \frac{1}{3}*0^3] )[/tex]
[tex]E(Y) =\frac{1}{100}[100*10 - 10*10^2 + \frac{1}{3}*10^3][/tex]
[tex]E(Y) =10 - 10 + \frac{1}{3}*10[/tex]
[tex]E(Y) =\frac{10}{3}[/tex]
Recall that:
[tex]Cov(X,Y) = E(XY) - E(X) \cdot E(Y)[/tex]
[tex]Cov(X,Y) = \frac{25}{3} - \frac{10}{3}*\frac{10}{3}[/tex]
[tex]Cov(X,Y) = \frac{25}{3} - \frac{100}{9}[/tex]
Take LCM
[tex]Cov(X,Y) = \frac{75- 100}{9}[/tex]
[tex]Cov(X,Y) = -\frac{ 25}{9}[/tex]
Related Questions
Marina spent 1/5 of her money on a book. If she had $80 left after she brought the book, how much money did she have originally?
Answers
Answer:
Marina had 50$ originally
Step-by-step explanation:
80-1/5 of the money she spent for the books that she brought, The step by step process is how much money she had left behind from the books she brought seperate
Answer:
$100
Step-by-step explanation:
Step 1, setting up an equation:
If Marina spent [tex]\frac{1}{5}[/tex] of her money, she has [tex]\frac{4}{5}[/tex] of her money remaining. We also are given that she has $80 left. We can set her initial total amount of money as x to solve this problem:
[tex]\frac{4}{5}x=80[/tex]
Step 2, solving for x:
[tex]\frac{4}{5}x=80\\\\4x=400\\\\x=100[/tex]
We have set x as her original amount of money, so we have the answer!
Marina originally had [tex]\fbox{100}[/tex] dollars.
I hope this helps! Let me know if you have any questions :)
The coeffcient of determination is the ratio of the explained variation to the total variation. A. The coefficient of determination is the ratio of the unexplained variation to the total variation. B. The coefficient of determination is the ratio of the total variation to the unexplained variation. C. The coefficient of determination is the ratio of the total variation to the explained variation. D. The coefficient of determination is the ratio of the explained variation to the total variation. Choose the correct answer below. A. The coefficient of determination is a measure of how closely two variables vary together. B. The coefficient of determination is the percent of the variation that is unexplained. C. The coefficient of determination is the percent of variation of y that is explained by the relationship between x and y. D. The coefficient of determination is the percent of the predicted values that equal the actual data values. Choose the correct answer below. A. The value 1r is a measure of how closely two variables vary together. B. The value 1r is the percent of the predicted values that equal the actual data values. C. The value 1r is the percent of the variation that is unexplained. D. The value 1r is the percent of variation of y that is explained by the relationship between x and y.
Answers
Answer:
D. The coefficient of determination is the ratio of the explained variation to the total variation.
C. The coefficient of determination is the percent of variation of y that is explained by the relationship between x and y
Step-by-step explanation:
The Coefficient of determination is the ratio of the explained variation in y to the total variation in y.
Coefficient of determination, r² ;
r² = explained variation / total variation
The coefficient of determination is the proportion of variation in y (predicted value) which is due to the regression line ; Hence, it is the percent of variation of y that is explained by the relationship between x and y.
Find a mathematical model for the below statement. (Use k for the constant of proportionality.) The gravitational attraction F between two objects of masses m1 and m2 is jointly proportional to the masses and inversely proportional to the square of the distance r between the objects.
Answers
Answer:
[tex]F = k\frac{m_1m_2}{r^2}[/tex]
Step-by-step explanation:
Direct proportional:
Multiplication
Inverse proportional:
Division
The gravitational attraction F between two objects of masses m1 and m2 is jointly proportional to the masses
This means that:
[tex]F = k\frac{m_1m_2}{}[/tex]
Inversely proportional to the square of the distance r between the objects.
Denominator is [tex]r^2[/tex]. Thus:
[tex]F = k\frac{m_1m_2}{r^2}[/tex]
the area of a square is 196m2. What is the length of one side
Answers
Answer:
14
Step-by-step explanation:
Simply take the square root of 196.
Hope that this helps!
Answer:
The side length is 14 m
Step-by-step explanation:
The formula for the area of a square is A = s^2, where s represents the length of one side.
If A = s^2 = 196m^2, we find the side length by taking the square root of both sides:
s = √(196 m^2) = 14 m
The side length is 14 m
The two way table shows information about preferred drinks of some people how many males drank only coffee
Answers
Answer:
73% The two-way table shows information about the preferred drinks of some people. b a How many males drank only coffee? b What is the probability that any person is male and only drinks coffee? 73% The two-way table shows information about the preferred drinks of some people
Find the equation of the line with Slope = 3 and passing through (4, 10) .write your equation in the form y=mx+b
Answers
Y=mx+b
Y=3x+b
Now replace the points x and y to find b (y-intercept)
10=3(4)+b
10-12=b
b= -2
Final answer:
Y=3x-2
Hope that answers your question and best of wishes!!
Data with 950 observations are drawn from a bell-shaped distribution with a mean of 78 and a standard deviation of 19. Approximately how many observations are more than 116
Answers
Answer: there are 22 observations that are more than 116.
Step-by-step explanation:
Let x be the random variable.
Given: Sample size = 950
Mean = 78
Standard deviation = 19
The probability that the observations are more than 116 is given by :-
[tex]P(X>116)=P(\dfrac{X-mean}{standard \ deviation}>\dfrac{116-78}{19})\\\\=P(Z>2)\\\\=1-P(Z<2)\\\\=1-0.9772\ \ \ [\text{Using p-value calculator}]\\\\=0.0228[/tex]
Number of observations are more than 116 = [tex]0.0228\times950[/tex]
[tex]=21.66\approx22[/tex]
Hence, there are 22 observations that are more than 116.
Evaluate -2yx - 4y what's the answer
Answers
Answer:
-2y (x+2)
Step-by-step explanation:
remove greatest common factors.
both terms have a (y) and a (-2)
Answer:
-2y ( x - 2 )
Step-by-step explanation:
- 2yx - 4y
factor out -2y from the equation
-2y ( x - 2 )
Joey has 2/3 of a gallon of ice cream. He divides the ice cream for his friends so that 1/9 of a gallon is in each bowl. How many bowls will he use?
A. 6
B. 1/6
C. 27
D. 1/9
Answers
Answer:
Joey will use 6 bowls.
Step-by-step explanation:
We have to divide to solve.
What is [tex]\frac{2}{3}[/tex] ÷ [tex]\frac{1}{9}[/tex] ?
When you divide fractions you just multiply by the reciprocal (the fraction flipped.)
[tex]\frac{2}{3}[/tex] ÷ [tex]\frac{1}{9}[/tex] = [tex]\frac{2}{3}[/tex] × [tex]\frac{9}{1}[/tex]
[tex]\frac{2}{3}[/tex] × [tex]\frac{9}{1}[/tex] = [tex]\frac{18}{3}[/tex]
[tex]\frac{18}{3}[/tex] = 6
Joey will use 6 bowls.
100 + 50
i'll date you if you answer this;)
Answers
Answer:
150
Step-by-step explanation:
100 + 50 = 150
ez
Q- getting a number greater than 3 or an odd number?
Answers
Step-by-step explanation:
So one's chance of getting a number greater than 3 is 3 chances out of 6 numbers that could be rolled or 1/2. Then we determine what the chances are of getting an ODD number. That includes 1, 3, & 5. So one's chance of getting an odd number is 3 chances out of the 6 numbers on the die, which is also 1/2.
Consider function g.
6,
-8 < x <-2 2
g(x) =
0,
-2
-4, 4 < x < 10
What are the values of the function when x = -2 and when r = 4?
gl-2)
=
3
g(4)
=
N
Answers
[tex]x = - 2 \: then \: g( - 2) = 0 \\ x = 4 \: then \: g(4) = - 4.[/tex]
Hence, The values of the function when x = -2 is g(x) = 0 and x = 4 is
g(x) = - 4.
What is composite function?A composite function is created when one function is substituted into another function.
Given
Composite Function g
g(x) = 6, -8 ≤ x ≤ -2
g(x) = 0, -2 ≤ x ≤ 4
g(x) = -4, 4 ≤ x ≤ 10
The values of the function when x = -2
x is between -2 and 4
g(x) = 0, -2 ≤ x ≤ 4
Thus, The values of the function when x = -2 is g(x) = 0
The values of the function when x = 4
x is between 4 and 10
g(x) = -4, 4 ≤ x ≤ 10
Thus, The values of the function when x = 4 is g(x) = - 4
Hence, The values of the function when x = -2 is g(x) = 0 and x = 4 is
g(x) = - 4.
Learn more about composite function here
https://brainly.com/question/20379727
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ASAP!!! Use the relationships in the diagrams below to calculate the the values of the variables if possible part. B
Answers
Answer: y = 57°;
Angle Values: 66° and 114°.
Step-by-step explanation:
The sum of the corresponding angles is 180°.
y + 9 + 2y = 180
3y = 180 - 9
3y = 171
y = 57
Angle Values:
57° + 9° = 66°
57° · 2 = 114°
..A tin of paint was 2/3 litres full. Tom used 1/2 of
the paint to paint his table. How much was left ?
Answers
Multiply the amount of paint started with by the amount used:
2/3 x 1/2 = (2 x 1) / (3 x 2) = 2/6 = 1/3
There is 1/3 liter left
on a tv game show,omar needs answer 7 out of every 10 questions correctly. There will be 30 questions in all
Answers
∫▒〖e^2x dx〗= ???? thanks
Answers
Answer:
Step-by-step explanation:
I=∫e^2x dx
put 2x=t
2dx=dt
dx=dt/2
I=1/2∫e^t dt
=1/2 e^t+c
=1/2 e^{2x}+c
You currently have a balance of $34 in your checking account. You then write a check for $76 dollars to pay for groceries. Finally, you deposit a check for $30.
What is your new balance?
Answers
Answer:
Correct me if I'm wrong but I'm sure it's -12
Step-by-step explanation:
34+30= 64
64-76= -12
Thể tích của khối cầu có bán kính 3a bằng
Answers
Answer:
[tex]36\pi {a}^{3} [/tex]
Step-by-step explanation:
Volume = [tex] \frac{4}{3} \times \pi \times (r)^{3} [/tex]
[tex] \frac{4}{3} \times \pi \times {3a}^{3} [/tex]
[tex] \frac{4\pi}{3} \times {3}^{3} [/tex]
[tex] \frac{4\pi}{3} \times 27[/tex]
[tex] \frac{4\pi}{3} \times (3(9))[/tex]
[tex]4\pi \times 9[/tex]
[tex]36\pi {a}^{3} [/tex]
Hope it is helpful...A business is interested in employees’ job satisfaction. The regional manager places a name card for every employee into a bowl and randomly selects 10 cards. Which sampling method was used?
cluster sampling
simple random sampling
stratified random sampling
systematic random sampling
Answers
Answer:
B
Step-by-step explanation:
Answer: simple random sampling
AKA: B
Step-by-step explanation: guy above me is correct :)
Pa help po plssssss
Sana po meron Ang maka-sagot ☺️☺️
Answers
Answer:
The answer is below
Step-by-step explanation:
a) the volume of the pit = length * width * depth = 9 dm * 6 dm * 7 dm = 378 dm³
Hence to fil the pit, 378 dm³ of sand is needed.
b) Volume of the room = length * width * height = 2.3 m * 0.59 m * 3.74 m = 5.08 m³
The cabinet would occupy 5.08 m³ of space.
c) volume = length * width * height
210 dm³ = 3.5 dm * 6 dm * width
width = 210 dm³ / (3.5 dm * 6 dm) = 10 dm
d) Volume of the room = length * width * height = 15 m * 15 m * 15 m = 3375 m³
The stockroom has a space of 3375 m³
e) volume = length * width * height = 5 dm * 5 dm * 5 dm = 125 dm³
The container has a space of 125 dm³
f) volume = length * width * height = 41 cm * 13 cm * 24 cm = 12792 cm³
The box has a space of 12792 cm³
g) volume of prism = length * width * height
72 m³ = 4 m * 3 m * length
length = 72 m³ / (4 m * 3 m) = 6 m
h) volume = length * width * height = 8 cm * 8 cm * 8 cm = 512 cm³
The figure has a volume of 512 cm³
i) volume = length * width * height = 5 ft * 2 ft * 3 ft = 30 ft³
The cabinet has a space of 30 ft³
Show number that 0.245 is Rational number
Answers
Answer:
A rational number is one that may be represented in the manner p/q.
245/1000 = 0.245
Step-by-step explanation:
Helppppp which one is it?
Answers
If a franchise company wanted to determine why sales were higher at some locations rather than others, what statistical process would be most appropriate? Group of answer choices Regression with sales as an independent variable Regression with sales as the dependent variable Use a confidence interval with the posted speed as the mean Hypothesis testing with a null hypothesis that the sales are less than or equal to the highest sales Hypothesis testing with a null hypothesis that the sales are more than the highest sales
Answers
Answer: Regression with sales as the dependent variable
Step-by-step explanation:
Since the franchise company wanted to determine why sales were higher at some locations rather than others, the statistical process that would be most appropriate is regression with sales as the dependent variable.
Regression will be used in determining the strength of the relationship that exist between one dependent variable and the independent variables. In this case, the dependent variable is sales.
HW HELP ASAP PLZZZZZ
Answers
Answer:
3
Step-by-step explanation:
I think its 3 because when we add 2 to x the graphic must be continue two step to right
Which expression can be modeled using the number line?
Answers
Answer:
7 x 1/10 = 7/10
Step-by-step explanation:
There are 7 "bumps" on the number line, and each "bump" moves the line 1/10.
This can also be modeled as
1/10 + 1/10 + 1/10 +1/10 +1/10 +1/10 +1/10 = 7/10
Your aunt is asking your help in figuring out how to
maximize profit for her pottery business. She makes
$35 on a bowl and $30 on a plate. She has just
ordered 40 pounds of clay. A bowl uses 5 pounds of
clay and a plate uses 4 pounds of clay. She already
has orders to make 4 bowls. How many bowls and
plates should she make in order to maximize her profit?
Step 1: Identify your variables and write the objective
function
What does x represent? What does y represent?
Answers
Answer:
650
Step-by-step explanation:
Một hộp đựng 10 phong bì bốc thư trúng thưởng trong đó 3 phong bì đựng 500 nghìn 7 phong bì đựng 100 nghìn. Bốc ngẫu nhiên liên tiếp hai phong bì. Nếu biết phong bì thứ hai có 500 nghìn, tìm xác xuất để phong bì đầu tiên cũng có 500 nghìn?
Answers
I can't understand tell me in english
Can someone help with this
Answers
Answer: 28
Step-by-step explanation: ADB = 1/2 ACB
HELP MATH ANSWER QUESTION
Answers
the answer is in the picture
given 12 consecutive integers in how many ways can three of these integers be selected to give a sum which divides by 4?
Answers
There are [tex]55[/tex] ways of these integers be selected to give a sum.
Given that:
It has [tex]12[/tex] consecutive integers .
Now,
Definition of consecutive integers :
" Consecutive integers are the integers that follow in the fixed sequence .
Consecutive integers is represented by [tex]n,n+1,n+2,n+3,....[/tex] where [tex]n[/tex] is an integer."
By given :
we have [tex]12[/tex] consecutive integers .
Thus,[tex]n=1[/tex] and substitute the equation is,
[tex]1,(n+1),(1+2),(1+3),(1+4),(1+5),(1+6),(1+7),(1+8),(1+9),(1+10),(1+11)\\\\\implies 1,2,3,4,5,6,7,8,9,10,11,12[/tex]
Now,
Split all the integers into 4 equal parts,
Part 1: Those integers are divisible by [tex]4[/tex] and the remainder be 0.
Then,
[tex]a=0(mod 4)[/tex]
Part 2: Those integer producing the remainder [tex]1[/tex] when it is divisible by [tex]4[/tex].
Then,
[tex]a=1(mod 4)[/tex]
Part 3: Those integer producing the remainder [tex]2[/tex] when it is divisible by[tex]4[/tex].
Then,
[tex]a=2(mod 4)[/tex]
Part 4: Those integer producing the remainder [tex]3[/tex] when it is divisible by[tex]4[/tex].
Then,
[tex]a=3(mod4)[/tex]
Since, three of these integers be selected to give a sum which divides by [tex]4[/tex] is,
In any 12 consecutive integers there are [tex]12\div4[/tex]
i.e. exactly 3 numbers of each category of mod4 namely ,
[tex]0(mod4), 1(mod4), 2(mod4), 3(mod4)[/tex]
Thus, the total combinations of above [tex]5[/tex] categories of sets of [tex]3[/tex] integers are
All the 3 numbers are [tex]0(mod4)[/tex][tex]3C_1=3(1)=3[/tex]
One number be [tex]0(mod4)[/tex] and other two numbers are [tex]2(mod4)[/tex][tex]3C_1*3C_2=3(1)*3=9[/tex]
One number [tex]0(mod4)[/tex] and other numbers [tex]1(mod4)[/tex] & [tex]3(mod4)[/tex][tex]3C_1*3C_1*3C_1=3*3*3=27[/tex]
Two numbers be [tex]1(mod4)[/tex] and one number be [tex]2(mod4)[/tex][tex]3C2 * 3C1 = 3*3 = 9[/tex]
One number be [tex]2(mod4)[/tex] and two numbers be [tex]3(mod4)[/tex][tex]3C1 * 3C2 = 3*3 = 9[/tex]
Thus, sum of the total ways be [tex]12[/tex] consecutive integers of three integers is divisible by 4 is,
[tex]3+9+27+9+9=55[/tex]
Hence, it has [tex]55[/tex] ways.
For more information,
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