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Mariulka [41]
3 years ago
11

Find the coordinates of the vertices of the triangle after a 270° rotation

Mathematics
1 answer:
Minchanka [31]3 years ago
8 0

Answer:

Step-by-step explanation:

(a). 270° clockwise or 90° counterclockwise:

(x, y) --------> ( - y, x )

(b). 270° counterclockwise or 90° clockwise:

(x, y) ---------> ( y, - x )

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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
Plzz helppp it's due today​
juin [17]

Answer:

8

Step-by-step explanation:

use trig functions

2 sides and 1 angle given

Sides given

  • hypotenuse -> x
  • opposite side to 30˚ -> 4

Angle

  • 30˚

trig function associated with hyp and opp is sin

sin(x) = opp/hyp

substitute

sin 30˚ = 4/x

solve for x

0.5 = 4/x

0.5x = 4

x = 8

7 0
3 years ago
Read 2 more answers
Plz help 10 points ;-;
Oksana_A [137]

Answer:

See explanation.

Step-by-step explanation:

Coordinates of point P is (2.5, 0).

Coordinates of point T are (-1.5, -1).

Coordinates of point R are (0.5, -2.5).

hope this helps.

5 0
3 years ago
Read 2 more answers
4x10^4 in standard notation
lyudmila [28]

Answer:

40000

Step-by-step explanation:

4×10^4 = 4×10000 = 40000

6 0
3 years ago
Read 2 more answers
Which of the following best describes the equation below?
Reika [66]
4(x + 9) + 5x = 9x - 36

=> 4x + 36 + 5x = 9x - 36

=> 9x + 36 = 9x - 36

=> 9x - 9x = -36 - 36

=> 0 = -72

You can see that the equation doesn't hold true.

So there is no solution for this equation.

Answer is C. no real solutions
5 0
3 years ago
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