This is the question. Below is the answer. The first line is
rather confusing. Please explain why that is. Why is it x2 instead
of x1.
How were the A_i"s chosen and why is there no contribution from
A
4.20 X₁ and X₂ are independent n(0, o²) random variables. (a) Find the joint distribution of Y₁ and Y2, where Y₁ = X² + X² and Y₂ = X₁ √vi (b) Show that Y₁ and Y₂ are independent,
Answers
The joint distribution of Y1 and Y2 is: P (Y1 ≤ y1, Y2 ≤ y2) = (1/2^(3/2)πσ^2V) ∫ [0 to y1/2] x ^ (1/2) exp (-x/2β) dx. Since, their joint distribution factorizes, Y1 and Y2 are independent.
To find the joint distribution of Y1 and Y2, we will first evaluate the expressions for Y1 and Y2. We have:
Y1 = X1² + X2²Y2 = X1√(V),
where X1 and X2 are independent N (0, σ^2) random variables.
Hence, we can write the joint distribution of Y1 and Y2 as:
P (Y1 ≤ y1, Y2 ≤ y2) = P [X1² + X2² ≤ y1, X1√(V) ≤ y2].
Now, we can express this in terms of X1 and X2 by using the transformation method. This involves computing the Jacobian, which is given by:
|J| = 2x2√(V).
After applying the transformation, we get:
P (Y1 ≤ y1, Y2 ≤ y2) = ∫∫ [f (x1, x2) |J|] dx1 dx2,
where f (x1, x2) is the joint probability density function of X1 and X2.
Since X1 and X2 are independent, we have:
f (x1, x2) = f (x1) * f (x2) = [1/(2πσ²)] exp (-x1²/2σ²) x [1/(2πσ²)] exp (-x2²/2σ²).
Therefore, the joint probability density function of Y1 and Y2 is:
P (Y1 ≤ y1, Y2 ≤ y2) = ∫∫[(1/4π²σ⁴) exp (-(x1²+x2²)/2σ²) x2√(V)] dx1dx2.
The integral can be simplified by making use of polar coordinates. We get:
P (Y1 ≤ y1, Y2 ≤ y2) = (1/4π²σ⁴) ∫ [0 to 2π] ∫ [0 to ∞] exp(-r²/2σ²) r√(V) drdθ.
Integrating over θ, we get:
P (Y1 ≤ y1, Y2 ≤ y2) = (1/2πσ⁴) ∫ [0 to ∞] exp(-r²/2σ²) r√(V) dr.
Integrating by parts, we get:
P (Y1 ≤ y1, Y2 ≤ y2) = (1/2σ⁴) ∫ [0 to ∞] exp(-r²/2σ²) (r²/2) V-1/2 dr.
This is a gamma distribution with parameters α = 1/2 and β = 1/2σ^2V. Therefore, the joint distribution of Y1 and Y2 is:
P (Y1 ≤ y1, Y2 ≤ y2) = (1/2^(3/2)πσ^2V) ∫ [0 to y1/2] x ^ (1/2) exp (-x/2β) dx.
To show that Y1 and Y2 are independent, we need to compute their marginal distributions and demonstrate that their joint distribution factorizes. This is a normal distribution with mean 0 and variance V. Hence, the joint distribution of Y1 and Y2 factorizes as:
P (Y1 ≤ y1, Y2 ≤ y2) = P (Y1 ≤ y1) * P (Y2 ≤ y2).
Therefore, Y1 and Y2 are independent.
Learn more about joint distribution visit:
brainly.com/question/14310262
#SPJ11
Related Questions
What is the value of I-9I?
Answers
Answer: 90
Step-by-step explanation:
91 - 1 = 90
Answer: 90
Step-by-step explanation:
Please help asap! q = ___ units
Answers
The length of q is 16 units in the given triangle.
Explain about substitution method ?
The substitution method is a technique for solving systems of equations in algebra. A system of equations is a set of two or more equations that must be solved simultaneously. The substitution method involves solving one equation for one variable, and then substituting the resulting expression into the other equation to eliminate that variable.
Here are the general steps for solving a system of equations using the substitution method:
According to the question :
To solve for q, Lets us assume a triangles ABC and ACD are similar. Therefore, we can set up the following proportion:
AC/AD = AB/AC
Substituting the given values, we get:
36/AD = 16/36
Multiplying both sides by AD and simplifying, we get:
AD = (36*16)/36 = 16 units
Therefore, the length of q is 16 units.
To know more about substitution method visit:
https://brainly.com/question/26743114
#SPJ1
PLSSS HELP GIVING BRAINLIEST
What is the circumference of a circle with a diameter of 7 feet? Use 3.14 for d = 7 ft O A. 21.98 ft O B. 38.47 ft O c. 43.96 ft O D. 10.99 ft
Answers
Answer:
i think it may be 21.98 but i dunno
Step-by-step explanation:
Find the product of these complex numbers.
(5+6i)(3-3i) =
A. 33 + 3i
B. 15 + 18i
C. 15 - 18i
D. -3 + 3i
Answers
Answer:
A) 33+3i
Step-by-step explanation:
Use the FOIL method: (a+b)(c+d)=ac+ad+bc+bd(a+b)(c+d)=ac+ad+bc+bd.
15−15i+18i+18
combine like terms
(15+18)+(−15i+18i)
simplify
33+3i
Charles and Jaden share a reward of $63 in a ratio of 2:5. How much does Jaden
get?
Answers
Answer:
9
Step-by-step explanation:
2:5=7
therefore Jaden got 63÷7
=9
PLEASE EXPLAIN YOUR ANSWER!
Answers
Answer:
B
Step-by-step explanation:
Basically you are dividing X by 4, meaning the number of boxes, if you took all the mugs out of the boxes there are 55 mugs.
see if you can find x ??
Answers
Answer:
just do 130 +5x+19=180
Answer:
Step-by-step explanation:
180° = 130°+5x+10
180 = 140+5X
-140 -140
40 = 5X
/5 /5
8 = X
Write a function in any form that would match the graph shown below.
Answers
Answer:
it has two zeros because it is touching x axis two times
Justin is making a snack. He toasts a piece of bread and spreads peanut butter and jelly on it. Then he cuts an apple into several slices and adds these to his plate. Next, Justin mixes a scoop of lemonade powder into a glass of water, mixes it, and adds ice.
How many chemical changes occurred in this scenario?
A.one
B.two
C.three
D.four
Answers
Answer:
One chemical change- It was when he toasted the bread. The heat changed the bread so that it has the crust on the outside.
Step-by-step explanation:
Hope it helps :)
Have a good day/night. Brainliest pls?
The dimensions of a screen are 21 ft by 20 ft. Find the length of the diagonal .
Answers
Answer:
29 ft
Step-by-step explanation:
the screens have 90 degree angles so they form right triangles with the diagonal, so use Pythagorean theorem to solve for the diagonal
diagonal² = 21² +20² , square and square root both sides of the equation
diagonal = √441+400, add
diagonal = √841, square root
diagonal = 29ft
which of the following statistics are unbiased estimators: a) all these (sample mean, sample proportion, and sample variance) b) the sample proportion c) the sample variance d) the sample mean
Answers
All these (sample mean, sample proportion, and sample variance) are the statistics are unbiased estimators.
What is meant by unbiased estimator?The difference between an estimator's expected value and the actual value of the parameter under consideration is known as bias in statistics. Unbiased estimators or decision rules are referred to as unbiased. "Bias" is a statistical term for an estimator's objective property.An estimator's expected value must exactly match the population parameter's value in order for it to be considered impartial. The discrepancy between an estimator's expected value and the parameter value is known as bias. The estimator has bias if this difference is not zero.Given a straightforward random sample, the sample mean is an unbiased estimator of the population parameter because, over a large number of samples, it does not consistently overestimate or underestimate the true population mean.Learn more about unbiased estimator refer to :
https://brainly.com/question/30237613
#SPJ4
Two altitudes of a triangle have lengths $12$ and $15$. What is the longest possible integer length of the third altitude
Answers
Let ABC be the given triangle. We can construct two triangles PAB and PBC such that they share the same height from P to AB and P to BC, respectively. We can label the side lengths of PAB and PBC as x and y, respectively. The total area of the triangle ABC is the sum of the areas of PAB and PBC:
Area_ABC = Area_PAB + Area_PBC We can write the area of each of the sub-triangles in terms of x and y by using the formula for the area of a triangle: Area_PAB = (1/2)(12)(x) = 6xArea_PBC = (1/2)(15)(y) = (15/2)y Setting the areas equal to each other and solving for y yields: y = (4/5)x Substituting this into the equation for the area of PBC yields:
Area_PBC = (1/2)(15/2)x = (15/4)x The area of ABC can also be written in terms of x by using the formula: Area_ABC = (1/2)(AB)(PQ) = (1/2)(12)(PQ) + (1/2)(15)(PQ) = (9/2)(PQ) Setting the areas equal to each other yields:(9/2)(PQ) = 6x + (15/4)x(9/2)(PQ) = (33/4)x(9/2)(PQ)/(33/4) = x(6/11)PQ = x(6/11)Thus, we can see that the longest possible integer length of the third altitude is $\boxed{66}$.
To know more about triangle visit:
https://brainly.com/question/2773823
#SPJ11
Complete the table for y=1/2x-3
Answers
Tables are created by substitute a set of input values into the function to create outputs. The required table is as shown below
x | y
0 -3
1 -2.5
2 -2
Tables and valuesTables are created by substitute a set of input values into the function to create outputs
Using x = 0, 1 and 2 as the input values
Given the function
y = 1/2x - 3
If x = 0
y = 1/2(0) - 3
y = -3
If x = 1
y = 1/2(1) - 3
y = -2.5
If x = 2
y = 1/2(2) - 3
y = -2
Hence the required table is as shown below
x | y
0 -3
1 -2.5
2 -2
Learn more on tables and values here: https://brainly.com/question/12151322
#SPJ1
a 20 foot tree stands parallel to a 12 foot flagpole.a 28 foot wire is stretched between the top of the tree and the top of the flagpole.how far apart are the tree and the flagpole
Answers
the distance between the tree and the flagpole is approximately 23.29 feet.
To find the distance between the tree and flagpole, we can use the Pythagorean theorem, which states that the square of the distance between the two points (the hypotenuse of the right triangle) is equal to the sum of the squares of the other two sides.
In this case, the distance between the tree and flagpole (the hypotenuse of the triangle) is equal to the square root of (20^2 + 12^2) = square root of (400 + 144) = square root of 544.
So the distance between the tree and the flagpole is approximately 23.29 feet.
Learn more about Pythagorean Theorem here: brainly.com/question/28361847
#SPJ4
Given an exchange rate of 1.21 dollar/pound and an exchange rate
of 1.22 dollar/euro, what is the exchange rate of the euro/pound
expressed to four decimal places. (Please do not put in any
currency s
Answers
The exchange rate of the euro/pound expressed to four decimal places is 1.0100 euro/pound.
To find the exchange rate of the euro/pound, we can use the given exchange rates of dollar/pound and dollar/euro.
Let's denote the exchange rate of euro/pound as E.
Given:
Exchange rate of dollar/pound = 1.21 dollar/pound
Exchange rate of dollar/euro = 1.22 dollar/euro
To find the exchange rate of euro/pound, we can divide the exchange rate of dollar/euro by the exchange rate of dollar/pound:
E = (Exchange rate of dollar/euro) / (Exchange rate of dollar/pound)
E = 1.22 dollar/euro / 1.21 dollar/pound
Simplifying this expression, we get:
E = 1.01 euro/pound
Therefore, the exchange rate of the euro/pound, is 1.0100 euro/pound.
To know more about rate refer here:
https://brainly.com/question/30354032#
#SPJ11
what am i doing wrong? i cannot figure out why i can't get the surface area. the answer says it's 69.3cm^3. please help me
Answers
Step-by-step explanation:
First, let's calculate surface area of cuboid first:
L = 4.8 cm
B = 1.5 cm
H = 2.1 cm
Surface Area of Cuboid = 2 x (LB + BH + LH)
= 2 x (4.8x1.5 + 1.5x2.1 + 4.8x2.1)
= 2 x ( 7.2 + 3.6 + 10.08)
= 41.76 cm2
Surface Area of Triangular prism = Area of Front and Back triangle + Area of Side triangles + Area of the bottom rectangle
Area of front and back triangles = 2 x (1/2)x 4.8 x 5.5 = 26.4 cm2
Area of side triangles = 2 x (1/2) x 1.5 x 5.5 = 8.25 cm2
Area of botton rectangle = 4.8 x 1.5 = 7.2 cm2
Now to get the area of the whole shape = Surface Area of Cuboid + Surface Area of Triangular prism - Area of the bottom rectangle
Note: In order to get the surface area of the given shape, the bottom of the prism or the top of the cuboid needs to be subtracted.
Area = 41.76 + (26.4+8.25) - 7.2
Area = 69.21 cm2
if the work required to stretch a spring 1 ft beyond its natural length is 15 ft-lb, how much work is needed to stretch it 6 in. beyond its natural length?
Answers
To stretch a spring 1 ft. from its natural length, a 15 ft-lb work is needed. To stretch a spring 6 in. from its natural length, the required work is 3.75 ft-lb
The work done on a spring is given by the formula:
W = 1/2 . kx²
Where:
k = spring constant
x = spring displacement
From the formula, we know that the work is directly proportional to the square of displacement, or mathematically:
W ∝ x²
Therefore,
W₁ : W₂ = x₁² : x₂²
Data from the problem
W₁ = 15 ft-lb
x₁ = 1 ft
x₂ = 6 in. = 0.5 ft
Hence,
15 : W₂ = 1² : 0.5²
W₂ = 0.25 x 15 = 3.75 ft-lb
Hence, the required work to stretch the spring 6 in beyond its natural length is 3.75 ft-lb
Learn more about spring here:
https://brainly.com/question/24339837
#SPJ4
The slope of a graph
Answers
Answer:
what are you asking?
Step-by-step explanation:
Slope formula is: (y2 minus y1) / (x2 minus x1)
Your nephew is four feet off the ground. He tosses his toy up into the air with initial velocity of 7 feet per second. How high is the toy at 1 second
Answers
Answer: AYO i aint even got a Nephew. I got a niece but she a dog tho. but still. If you love itachi let me get this point. ありがとうございました ( thank you)
Step-by-step explanation:
describe how to estimate the regression function (equivalently, a,b) by the least-squares principle. notes: you only need to provide a description. this is an example of a nonlinear regression model.
Answers
The estimates are used to construct the estimated regression equation:
ŷ = b0 + b1x.
What is the estimation of the regression function?Regression analysis is typically performed for one of two reasons: to estimate the influence of some explanatory variable on the dependent variable or to predict the value of the dependent variable for individuals for whom some information about the explanatory variables is available. Y = mX + b is the equation for simple linear regression, where X is the predictor (independent) variable, m is the estimated slope, and b is the estimated intercept. An estimated regression equation is created using these estimates: ŷ = b0 + b1xA straight line is a good approximation to the relationship between y and x on the graph of the estimated regression equation for simple linear regression.Therefore, the estimates are used to construct the estimated regression equation: ŷ = b0 + b1x.
Know more about regression function here:
https://brainly.com/question/25987747
#SPJ9
Grapes cost $2.39 per pound. Bananas cost $0.59 per pound. You have $15 to spend on g pounds of grapes and b pounds of bananas.
Answers
Answer:
2.39g + 0.59b = 15
Step-by-step explanation:
Algebraic Expressions
Let:
g = pounds of grapes
b = pounds of bananas
Since grapes cost $2.39 per pound, g pounds of grapes cost 2.39g dollars.
Since bananas cost $0.59 per pound, b pounds of bananas cost 0.59b dollars.
The total cost of fruits is:
C = 2.39g + 0.59b
And this cost is equal to $15:
2.39g + 0.59b = 15
This is the equation that relates the costs of the fruits. If another condition was given, then we could find how many pounds of each fruit is to be purchased.
4 What transformation will the following function undergo? Click all that apply?y =- 2(x + 8)? – 5A) vertical stretch B) vertical compression C) reflection across the x-axis D) translated to the left 8 units E) translated to the right 8 units F) translated up 5 units G) translated down 5 units
Answers
Firstly, the function is shifted 5 units down because of the -5 at the end of the function. OPTION G
It is also translated 8 units to the left because +8 is added to x. OPTION D
The function is also vertically compressed. OPTION B
The graph will be depicted below:
Find missing side? Soh cah toa
Answers
Answer:
x ≈ 14.6
Step-by-step explanation:
Using the sine ratio in the right triangle
sin59° = \(\frac{opposite}{hypotenuse}\) = \(\frac{x}{17}\) ( multiply both sides by 17 )
17 × sin59° = x , then
x ≈ 14.6 ( to 1 dec. place )
In Tableau, a(n) ________ allows a visualization developer or
user to dynamically remove rows from the visualization
a) Filter
b) calculated field
c) parameter
d) table calculation
Answers
In Tableau, a filter allows a visualization developer or user to dynamically remove rows from the visualization. Thus, Option A is the answer.
What is Tableau?
Tableau is an interactive and intuitive data visualization tool that enables people to see and understand data. It provides a variety of data exploration, modelling, and visualization options. Tableau is one of the best data visualization software available in the market. It provides an easy way to integrate data from various sources. With Tableau, you can easily create interactive and visually appealing charts, graphs, and other types of visualizations that are easy to understand and interpret.
What is a filter?
In Tableau, a filter is a tool that allows you to selectively display data from a larger dataset based on certain conditions. Filters help you to narrow down your data to the most relevant information so you can make better decisions based on your analysis. Filters can be applied to different dimensions and measures to help you get a better view of your data. Tableau allows you to filter your data using various types of filters such as categorical, continuous, and relative filters. You can also create custom filters using calculated fields and parameters.
A filter allows a visualization developer or user to dynamically remove rows from the visualization. Therefore, the correct option is (a) Filter.
Learn more about "Tableau": https://brainly.com/question/29838450
#SPJ11
Anyone know the song you broke me first?
Answers
Answer:
Yes its very sad (the words)
Step-by-step explanation:
Not going to lie but it does kind of sound like camila cabeo sang it.
I really like the tune and the singing too
The volume of a cylinder is given by the formula V = ²h,
where r is the radius and h is the height of the cylinder.
Determine the volume of a soup can with a 3-inch radius and
a 5.5-inch height.
Use 3.14 for T.
a 145.51 in ³
b 245.51 in ³
c 103.62 in ³
d 155.43 in ³
Activate
Go to Settin
Answers
V= 3.14(3)^2•5.5
Now, solve with PEMDAS:
Exponents come first:
V=3.14(9)•5.5
Then, multiplication and division from left to right:
V=28.26•5.5
V=155.43
Therefore, the volume is 155.43in^3
-7 _ -4?
great than less than or what ASAP PLEASE!!!!
Answers
Answer:
-7 < -4
Step-by-step explanation:
can you do it by telling
Answers
Answer:
Step-byoooooooooooo
-step explanation:
0/6 is nothing but 0. Anything divided by zero is zero. Since A represents 0 on the number line, A would be the answer. :)
If the velocity of a seagull is 0 m/s for a given time period, what can you say about the position of the seagull at the end of the time period?
Answers
Answer:
that the seagull hasn't moved
Step-by-step explanation:
probably dead
what is the value of x/y when x = -15 and y = -3
A 5
B -5
C 12
D -12
Answers
Answer:
x= -15
y= -3
x/y
=-15/-3
=5
option A is correct
as dividing 15 by 3 it’ll be 5
and the negative signs when divided are canceled