Answer:
0
Step-by-step explanation:
everything multiplied by 0 is 0
Answer:
1 and 5
Step-by-step explanation:
So basically corresponding angles are the angles which lie on the same line of the transversal
And also one of them will be an interior angle and the other a exterior angle
The expected length of code for one encoded symbol is
![\displaystyle\sum_{\alpha\in\{w,x,y,z\}}p_\alpha\ell_\alpha](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Csum_%7B%5Calpha%5Cin%5C%7Bw%2Cx%2Cy%2Cz%5C%7D%7Dp_%5Calpha%5Cell_%5Calpha)
where
is the probability of picking the letter
, and
is the length of code needed to encode
.
is given to us, and we have
![\begin{cases}\ell_w=1\\\ell_x=2\\\ell_y=\ell_z=3\end{cases}](https://tex.z-dn.net/?f=%5Cbegin%7Bcases%7D%5Cell_w%3D1%5C%5C%5Cell_x%3D2%5C%5C%5Cell_y%3D%5Cell_z%3D3%5Cend%7Bcases%7D)
so that we expect a contribution of
![\dfrac12+\dfrac24+\dfrac{2\cdot3}8=\dfrac{11}8=1.375](https://tex.z-dn.net/?f=%5Cdfrac12%2B%5Cdfrac24%2B%5Cdfrac%7B2%5Ccdot3%7D8%3D%5Cdfrac%7B11%7D8%3D1.375)
bits to the code per encoded letter. For a string of length
, we would then expect
.
By definition of variance, we have
![\mathrm{Var}[L]=E\left[(L-E[L])^2\right]=E[L^2]-E[L]^2](https://tex.z-dn.net/?f=%5Cmathrm%7BVar%7D%5BL%5D%3DE%5Cleft%5B%28L-E%5BL%5D%29%5E2%5Cright%5D%3DE%5BL%5E2%5D-E%5BL%5D%5E2)
For a string consisting of one letter, we have
![\displaystyle\sum_{\alpha\in\{w,x,y,z\}}p_\alpha{\ell_\alpha}^2=\dfrac12+\dfrac{2^2}4+\dfrac{2\cdot3^2}8=\dfrac{15}4](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Csum_%7B%5Calpha%5Cin%5C%7Bw%2Cx%2Cy%2Cz%5C%7D%7Dp_%5Calpha%7B%5Cell_%5Calpha%7D%5E2%3D%5Cdfrac12%2B%5Cdfrac%7B2%5E2%7D4%2B%5Cdfrac%7B2%5Ccdot3%5E2%7D8%3D%5Cdfrac%7B15%7D4)
so that the variance for the length such a string is
![\dfrac{15}4-\left(\dfrac{11}8\right)^2=\dfrac{119}{64}\approx1.859](https://tex.z-dn.net/?f=%5Cdfrac%7B15%7D4-%5Cleft%28%5Cdfrac%7B11%7D8%5Cright%29%5E2%3D%5Cdfrac%7B119%7D%7B64%7D%5Capprox1.859)
"squared" bits per encoded letter. For a string of length
, we would get
.
The
-coordinate of the center of mass is the average value of
over the plate, given by the ratio of the integral of
to the mass of the plate.
The mass of the plate is
![\displaystyle \int_0^1 \int_{y^2}^1 xy \, dx \, dy](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint_0%5E1%20%5Cint_%7By%5E2%7D%5E1%20xy%20%5C%2C%20dx%20%5C%2C%20dy)
while the integral of
over the plate is
![\displaystyle \int_0^1 \int_{y^2}^1 xy^2 \, dx \, dy](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint_0%5E1%20%5Cint_%7By%5E2%7D%5E1%20xy%5E2%20%5C%2C%20dx%20%5C%2C%20dy)
and the
-coordinate of the center of mass is
![\boxed{\dfrac{\displaystyle \int_0^1 \int_{y^2}^1 xy^2 \, dx \, dy}{\displaystyle \int_0^1 \int_{y^2}^1 xy \, dx \, dy}}](https://tex.z-dn.net/?f=%5Cboxed%7B%5Cdfrac%7B%5Cdisplaystyle%20%5Cint_0%5E1%20%5Cint_%7By%5E2%7D%5E1%20xy%5E2%20%5C%2C%20dx%20%5C%2C%20dy%7D%7B%5Cdisplaystyle%20%5Cint_0%5E1%20%5Cint_%7By%5E2%7D%5E1%20xy%20%5C%2C%20dx%20%5C%2C%20dy%7D%7D)