Answer:
Step-by-step explanation:
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Answer:
120
Step-by-step explanation:
Okay so the equation would be like this
30 + 0.10g = P
g = GB of data used
P = Total Price
Let's work this out now!
So. If his total bill was $42 lets subtract the 30 that comes out every month automatically, now we are just left with 12, if each GB costs 0.10 then all you would do to find out how many GB he used is do 12/0.10 (which is really just 12*10) which will get you 120
Meaning that he used 120 GB
(That's a lot!)
I'll leave the computation via R to you. The 
are distributed uniformly on the intervals ![[0,10i]](https://tex.z-dn.net/?f=%5B0%2C10i%5D)
, so that
each with mean/expectation
![E[W_i]=\displaystyle\int_{-\infty}^\infty wf_{W_i}(w)\,\mathrm dw=\int_0^{10i}\frac w{10i}\,\mathrm dw=5i](https://tex.z-dn.net/?f=E%5BW_i%5D%3D%5Cdisplaystyle%5Cint_%7B-%5Cinfty%7D%5E%5Cinfty%20wf_%7BW_i%7D%28w%29%5C%2C%5Cmathrm%20dw%3D%5Cint_0%5E%7B10i%7D%5Cfrac%20w%7B10i%7D%5C%2C%5Cmathrm%20dw%3D5i)
and variance
![\mathrm{Var}[W_i]=E[(W_i-E[W_i])^2]=E[{W_i}^2]-E[W_i]^2](https://tex.z-dn.net/?f=%5Cmathrm%7BVar%7D%5BW_i%5D%3DE%5B%28W_i-E%5BW_i%5D%29%5E2%5D%3DE%5B%7BW_i%7D%5E2%5D-E%5BW_i%5D%5E2)
We have
![E[{W_i}^2]=\displaystyle\int_{-\infty}^\infty w^2f_{W_i}(w)\,\mathrm dw=\int_0^{10i}\frac{w^2}{10i}\,\mathrm dw=\frac{100i^2}3](https://tex.z-dn.net/?f=E%5B%7BW_i%7D%5E2%5D%3D%5Cdisplaystyle%5Cint_%7B-%5Cinfty%7D%5E%5Cinfty%20w%5E2f_%7BW_i%7D%28w%29%5C%2C%5Cmathrm%20dw%3D%5Cint_0%5E%7B10i%7D%5Cfrac%7Bw%5E2%7D%7B10i%7D%5C%2C%5Cmathrm%20dw%3D%5Cfrac%7B100i%5E2%7D3)
so that
![\mathrm{Var}[W_i]=\dfrac{25i^2}3](https://tex.z-dn.net/?f=%5Cmathrm%7BVar%7D%5BW_i%5D%3D%5Cdfrac%7B25i%5E2%7D3)
Now,
![E[W_1+W_2+W_3]=E[W_1]+E[W_2]+E[W_3]=5+10+15=30](https://tex.z-dn.net/?f=E%5BW_1%2BW_2%2BW_3%5D%3DE%5BW_1%5D%2BE%5BW_2%5D%2BE%5BW_3%5D%3D5%2B10%2B15%3D30)
and
![\mathrm{Var}[W_1+W_2+W_3]=E\left[\big((W_1+W_2+W_3)-E[W_1+W_2+W_3]\big)^2\right]](https://tex.z-dn.net/?f=%5Cmathrm%7BVar%7D%5BW_1%2BW_2%2BW_3%5D%3DE%5Cleft%5B%5Cbig%28%28W_1%2BW_2%2BW_3%29-E%5BW_1%2BW_2%2BW_3%5D%5Cbig%29%5E2%5Cright%5D)
![\mathrm{Var}[W_1+W_2+W_3]=E[(W_1+W_2+W_3)^2]-E[W_1+W_2+W_3]^2](https://tex.z-dn.net/?f=%5Cmathrm%7BVar%7D%5BW_1%2BW_2%2BW_3%5D%3DE%5B%28W_1%2BW_2%2BW_3%29%5E2%5D-E%5BW_1%2BW_2%2BW_3%5D%5E2)
We have
![E[(W_1+W_2+W_3)^2]](https://tex.z-dn.net/?f=E%5B%28W_1%2BW_2%2BW_3%29%5E2%5D)
![=E[{W_1}^2]+E[{W_2}^2]+E[{W_3}^2]+2(E[W_1]E[W_2]+E[W_1]E[W_3]+E[W_2]E[W_3])](https://tex.z-dn.net/?f=%3DE%5B%7BW_1%7D%5E2%5D%2BE%5B%7BW_2%7D%5E2%5D%2BE%5B%7BW_3%7D%5E2%5D%2B2%28E%5BW_1%5DE%5BW_2%5D%2BE%5BW_1%5DE%5BW_3%5D%2BE%5BW_2%5DE%5BW_3%5D%29)
because and 
are independent when 
, and so
![E[(W_1+W_2+W_3)^2]=\dfrac{100}3+\dfrac{400}3+300+2(50+75+150)=\dfrac{3050}3](https://tex.z-dn.net/?f=E%5B%28W_1%2BW_2%2BW_3%29%5E2%5D%3D%5Cdfrac%7B100%7D3%2B%5Cdfrac%7B400%7D3%2B300%2B2%2850%2B75%2B150%29%3D%5Cdfrac%7B3050%7D3)
giving a variance of
![\mathrm{Var}[W_1+W_2+W_3]=\dfrac{3050}3-30^2=\dfrac{350}3](https://tex.z-dn.net/?f=%5Cmathrm%7BVar%7D%5BW_1%2BW_2%2BW_3%5D%3D%5Cdfrac%7B3050%7D3-30%5E2%3D%5Cdfrac%7B350%7D3)
and so the standard deviation is 
# # #
A faster way, assuming you know the variance of a linear combination of independent random variables, is to compute
![\mathrm{Var}[W_1+W_2+W_3]](https://tex.z-dn.net/?f=%5Cmathrm%7BVar%7D%5BW_1%2BW_2%2BW_3%5D)
![=\mathrm{Var}[W_1]+\mathrm{Var}[W_2]+\mathrm{Var}[W_3]+2(\mathrm{Cov}[W_1,W_2]+\mathrm{Cov}[W_1,W_3]+\mathrm{Cov}[W_2,W_3])](https://tex.z-dn.net/?f=%3D%5Cmathrm%7BVar%7D%5BW_1%5D%2B%5Cmathrm%7BVar%7D%5BW_2%5D%2B%5Cmathrm%7BVar%7D%5BW_3%5D%2B2%28%5Cmathrm%7BCov%7D%5BW_1%2CW_2%5D%2B%5Cmathrm%7BCov%7D%5BW_1%2CW_3%5D%2B%5Cmathrm%7BCov%7D%5BW_2%2CW_3%5D%29)
and since the are independent, each covariance is 0. Then
![\mathrm{Var}[W_1+W_2+W_3]=\mathrm{Var}[W_1]+\mathrm{Var}[W_2]+\mathrm{Var}[W_3]](https://tex.z-dn.net/?f=%5Cmathrm%7BVar%7D%5BW_1%2BW_2%2BW_3%5D%3D%5Cmathrm%7BVar%7D%5BW_1%5D%2B%5Cmathrm%7BVar%7D%5BW_2%5D%2B%5Cmathrm%7BVar%7D%5BW_3%5D)
![\mathrm{Var}[W_1+W_2+W_3]=\dfrac{25}3+\dfrac{100}3+75=\dfrac{350}3](https://tex.z-dn.net/?f=%5Cmathrm%7BVar%7D%5BW_1%2BW_2%2BW_3%5D%3D%5Cdfrac%7B25%7D3%2B%5Cdfrac%7B100%7D3%2B75%3D%5Cdfrac%7B350%7D3)
and take the square root to get the standard deviation.
Answer: ok i did :)
Step-by-step explanation: how are you?
