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2 years ago

8

Circle a has an area of 254.469 square units. What is the circumstance of circle a? Round your answer to the nearest hundredth,

if necessary.

1 answer:

Genrish500 [490]2 years ago

7 0

Sqrt(254.469/<span>π)=8.9999

</span>

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2) X and Y are jointly continuous with joint pdf

Nady [450]

From what I gather from your latest comments, the PDF is given to be

$f_{X,Y}(x,y)=\begin{cases}cxy&\text{for }0\le x,y \le1\\0&\text{otherwise}\end{cases}$

and in particular, <em>f(x, y)</em> = <em>cxy</em> over the unit square [0, 1]², meaning for 0 ≤ <em>x</em> ≤ 1 and 0 ≤ <em>y</em> ≤ 1. (As opposed to the unbounded domain, <em>x</em> ≤ 0 *and* <em>y</em> ≤ 1.)

(a) Find <em>c</em> such that <em>f</em> is a proper density function. This would require

$\displaystyle\int_0^1\int_0^1 cxy\,\mathrm dx\,\mathrm dy=c\left(\int_0^1x\,\mathrm dx\right)\left(\int_0^1y\,\mathrm dy\right)=\frac c{2^2}=1\implies \boxed{c=4}$

(b) Get the marginal density of <em>X</em> by integrating the joint density with respect to <em>y</em> :

$f_X(x)=\displaystyle\int_0^1 4xy\,\mathrm dy=(2xy^2)\bigg|_{y=0}^{y=1}=\begin{cases}2x&\text{for }0\le x\le 1\\0&\text{otherwise}\end{cases}$

(c) Get the marginal density of <em>Y</em> by integrating with respect to <em>x</em> instead:

$f_Y(y)=\displaystyle\int_0^14xy\,\mathrm dx=\begin{cases}2y&\text{for }0\le y\le1\\0&\text{otherwise}\end{cases}$

(d) The conditional distribution of <em>X</em> given <em>Y</em> can obtained by dividing the joint density by the marginal density of <em>Y</em> (which follows directly from the definition of conditional probability):

$f_{X\mid Y}(x\mid y)=\dfrac{f_{X,Y}(x,y)}{f_Y(y)}=\begin{cases}2x&\text{for }0\le x\le 1\\0&\text{otherwise}\end{cases}$

(e) From the definition of expectation:

$E[X]=\displaystyle\int_0^1\int_0^1 x\,f_{X,Y}(x,y)\,\mathrm dx\,\mathrm dy=4\left(\int_0^1x^2\,\mathrm dx\right)\left(\int_0^1y\,\mathrm dy\right)=\boxed{\frac23}$

$E[Y]=\displaystyle\int_0^1\int_0^1 y\,f_{X,Y}(x,y)\,\mathrm dx\,\mathrm dy=4\left(\int_0^1x\,\mathrm dx\right)\left(\int_0^1y^2\,\mathrm dy\right)=\boxed{\frac23}$

$E[XY]=\displaystyle\int_0^1\int_0^1xy\,f_{X,Y}(x,y)\,\mathrm dx\,\mathrm dy=4\left(\int_0^1x^2\,\mathrm dx\right)\left(\int_0^1y^2\,\mathrm dy\right)=\boxed{\frac49}$

(f) Note that the density of <em>X</em> | <em>Y</em> in part (d) identical to the marginal density of <em>X</em> found in (b), so yes, <em>X</em> and <em>Y</em> are indeed independent.

The result in (e) agrees with this conclusion, since E[<em>XY</em>] = E[<em>X</em>] E[<em>Y</em>] (but keep in mind that this is a property of independent random variables; equality alone does not imply independence.)

8 0

3 years ago

Write three fractions between 3/10 and 9/10

Norma-Jean [14]

Answer:

4/10, 5/10, 6/10 are all fractions in between 3/10 and 9/10

Step-by-step explanation:

8 0

3 years ago

What is the root of -9 ? Thank you for help

stiks02 [169]

I don't think this is possible...

But if you mean 9 then the root is 3

7 0

2 years ago

An 8-sided die is formed when two square pyramids are attached by their bases. The perimeter of the base shape is 6 centimeters.

lawyer [7]

Answer:

7.5cm²

Complete question:What is the surface area of the 8-sided figure?

Step-by-step explanation:

Consider the half of the 8-sided die i.e the pyramid ABCDE:

As the base is 6 of the perimeter, so each side of the base is 6/4=3/2

Now, the distance between the tallest points is 2 cm, so the height of Pyramid ABCDE is 1 cm.

Considering the right triangle EOM:

EO=1, OM= 3/4 (it is half of AB=3/2) and EM is the hypothenuse, so by the pythagorean theorem length of EM is

$\sqrt{1+ ( \frac{3}{4} )^{2} }= \sqrt{1+ \frac{9}{16} }= \sqrt{ \frac{25}{16} }= \frac{5}{4} (cm)$

5. So side EBC has area

$\frac{1}{2}BC*EM= \frac{1}{2}* \frac{3}{2}* \frac{5}{4}= \frac{15}{16}$

6. The total area is 8*Area(EBC)= $8* \frac{15}{16}= \frac{15}{2}=7.5 ( cm^{2} )$

3 0

2 years ago

What two numbers multiply to 6 but add up to -1?

Artyom0805 [142]

Answer:

-3 and 2

Step-by-step explanation:

7 0

2 years ago

Read 2 more answers