Answer: 
Mustafa could have written more than 8 articles for the newspaper.
Step-by-step explanation:
Given: Heloise has written 1/4 as many articles as Mustafa has. Gia has written 3/2 as many articles as Mustafa has.
Let m be the number of articles Mustafa writes.
Then articles written by Heloise=
And articles written by Gia=
The total number of articles =
Mustafa, Heloise, and Gia have written more than a combined total of 22 articles for the school newspaper.
Hence, Mustafa could have written more than 8 articles for the newspaper.
Answer: The approximate total weight of the grapefruits, using the clustering estimation technique is B. 35 ounces.
Given that the perimeter of a regular hexagon is 75 feet, determine the length of one side.
Given:
Perimeter : 75 ft.
Find:
Side Length:
Work:
Hexagon = 6 six sides
Divide perimeter by amount of sides (6).
75/6 = 12.5
1 side = 12.5 ft
Thus, the length of side on the hexagon is 12.5ft or 12 and 1/2 feet.
First, you'll need to find the marginal distributions of 
. By the law of total probability,
which translates to
Similarly,
Compute the expectations for both random variables:
![E[X]=\displaystyle\int_{-\infty}^\infty x\,f_X(x)\,\mathrm dx=\int_0^12x(1-x)\,\mathrm dx=\frac13](https://tex.z-dn.net/?f=E%5BX%5D%3D%5Cdisplaystyle%5Cint_%7B-%5Cinfty%7D%5E%5Cinfty%20x%5C%2Cf_X%28x%29%5C%2C%5Cmathrm%20dx%3D%5Cint_0%5E12x%281-x%29%5C%2C%5Cmathrm%20dx%3D%5Cfrac13)
![E[Y]=\displaystyle\int_{-\infty}^\infty y\,f_Y(y)\,\mathrm dy=\int_0^12y^2\,\mathrm dy=\frac23](https://tex.z-dn.net/?f=E%5BY%5D%3D%5Cdisplaystyle%5Cint_%7B-%5Cinfty%7D%5E%5Cinfty%20y%5C%2Cf_Y%28y%29%5C%2C%5Cmathrm%20dy%3D%5Cint_0%5E12y%5E2%5C%2C%5Cmathrm%20dy%3D%5Cfrac23)
Compute the variances and thus standard deviations:
![V[X]=E[(X-E[X])^2]=E[X^2]-E[X]^2](https://tex.z-dn.net/?f=V%5BX%5D%3DE%5B%28X-E%5BX%5D%29%5E2%5D%3DE%5BX%5E2%5D-E%5BX%5D%5E2)
where
![E[X^2]=\displaystyle\int_{-\infty}^\infty x^2\,f_X(x)\,\mathrm dx=\int_0^12x^2(1-x)\,\mathrm dx=\frac16](https://tex.z-dn.net/?f=E%5BX%5E2%5D%3D%5Cdisplaystyle%5Cint_%7B-%5Cinfty%7D%5E%5Cinfty%20x%5E2%5C%2Cf_X%28x%29%5C%2C%5Cmathrm%20dx%3D%5Cint_0%5E12x%5E2%281-x%29%5C%2C%5Cmathrm%20dx%3D%5Cfrac16)
![\implies V[X]=\dfrac16-\left(\dfrac13\right)^2=\dfrac1{18}\implies\sqrt{V[X]}=\dfrac1{3\sqrt2}](https://tex.z-dn.net/?f=%5Cimplies%20V%5BX%5D%3D%5Cdfrac16-%5Cleft%28%5Cdfrac13%5Cright%29%5E2%3D%5Cdfrac1%7B18%7D%5Cimplies%5Csqrt%7BV%5BX%5D%7D%3D%5Cdfrac1%7B3%5Csqrt2%7D)
![E[Y^2]=\displaystyle\int_{\infty}^\infty y^2f_Y(y)\,\mathrm dy=\int_0^12y^3\,\mathrm dy=\frac12](https://tex.z-dn.net/?f=E%5BY%5E2%5D%3D%5Cdisplaystyle%5Cint_%7B%5Cinfty%7D%5E%5Cinfty%20y%5E2f_Y%28y%29%5C%2C%5Cmathrm%20dy%3D%5Cint_0%5E12y%5E3%5C%2C%5Cmathrm%20dy%3D%5Cfrac12)
![\implies V[Y]=\dfrac12-\left(\dfrac23\right)^2=\dfrac1{18}\implies\sqrt{V[Y]}=\dfrac1{3\sqrt2}](https://tex.z-dn.net/?f=%5Cimplies%20V%5BY%5D%3D%5Cdfrac12-%5Cleft%28%5Cdfrac23%5Cright%29%5E2%3D%5Cdfrac1%7B18%7D%5Cimplies%5Csqrt%7BV%5BY%5D%7D%3D%5Cdfrac1%7B3%5Csqrt2%7D)
Compute the covariance:
![\operatorname{Cov}[X,Y]=E[(X-E[X])(Y-E[Y])]=E[XY]-E[X]E[Y]](https://tex.z-dn.net/?f=%5Coperatorname%7BCov%7D%5BX%2CY%5D%3DE%5B%28X-E%5BX%5D%29%28Y-E%5BY%5D%29%5D%3DE%5BXY%5D-E%5BX%5DE%5BY%5D)
We have
![E[XY]=\displaystyle\int_{-\infty}^\infty\int_{-\infty}^\infty xy\,f_{X,Y}(x,y)\,\mathrm dx\,\mathrm dy=\int_0^1\int_0^y2xy\,\mathrm dx\,\mathrm dy=\frac14](https://tex.z-dn.net/?f=E%5BXY%5D%3D%5Cdisplaystyle%5Cint_%7B-%5Cinfty%7D%5E%5Cinfty%5Cint_%7B-%5Cinfty%7D%5E%5Cinfty%20xy%5C%2Cf_%7BX%2CY%7D%28x%2Cy%29%5C%2C%5Cmathrm%20dx%5C%2C%5Cmathrm%20dy%3D%5Cint_0%5E1%5Cint_0%5Ey2xy%5C%2C%5Cmathrm%20dx%5C%2C%5Cmathrm%20dy%3D%5Cfrac14)
and so
![\operatorname{Cov}[X,Y]=\dfrac14-\dfrac13\dfrac23=\dfrac1{36}](https://tex.z-dn.net/?f=%5Coperatorname%7BCov%7D%5BX%2CY%5D%3D%5Cdfrac14-%5Cdfrac13%5Cdfrac23%3D%5Cdfrac1%7B36%7D)
Finally, the correlation:
![\operatorname{Corr}[X,Y]=\dfrac{\operatorname{Cov}[X,Y]}{\sqrt{V[X]}\sqrt{V[Y]}}=\dfrac{\frac1{36}}{\left(\frac1{3\sqrt2}\right)^2}=\dfrac12](https://tex.z-dn.net/?f=%5Coperatorname%7BCorr%7D%5BX%2CY%5D%3D%5Cdfrac%7B%5Coperatorname%7BCov%7D%5BX%2CY%5D%7D%7B%5Csqrt%7BV%5BX%5D%7D%5Csqrt%7BV%5BY%5D%7D%7D%3D%5Cdfrac%7B%5Cfrac1%7B36%7D%7D%7B%5Cleft%28%5Cfrac1%7B3%5Csqrt2%7D%5Cright%29%5E2%7D%3D%5Cdfrac12)
