Question

Suppose that W1, W2, and W3 are independent uniform random variables with the following distributions: Wi ~ Uni(0,10*i). What is the standard deviation of W1+W2+W3? Using R, compute this value. Round the answer to 2 decimal places and enter into the box:

Answer 1

I'll leave the computation via R to you. The
`W_i` are distributed uniformly on the intervals
`[0,10i]`, so that

`f_(W_i)(w)=\begin{cases}\frac1{10i}&\text{for }0\le w\le10i\\\\0&\text{otherwise}\end{cases}`

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`

and variance

`\mathrm{Var}[W_i]=E[(W_i-E[W_i])^2]=E[{W_i}^2]-E[W_i]^2`

We have

`E[{W_i}^2]=\displaystyle\int_(-\infty)^\infty w^2f_(W_i)(w)\,\mathrm dw=\int_0^(10i)(w^2)/(10i)\,\mathrm dw=\frac{100i^2}3`

so that

`\mathrm{Var}[W_i]=\frac{25i^2}3`

Now,

`E[W_1+W_2+W_3]=E[W_1]+E[W_2]+E[W_3]=5+10+15=30`

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]`

`\mathrm{Var}[W_1+W_2+W_3]=E[(W_1+W_2+W_3)^2]-E[W_1+W_2+W_3]^2`

We have

`(W_1+W_2+W_3)^2={W_1}^2+{W_2}^2+{W_3}^2+2(W_1W_2+W_1W_3+W_2W_3)`

`E[(W_1+W_2+W_3)^2]`

`=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])`

because
`W_i` and
`W_j` are independent when
`i\\eq j`, and so

`E[(W_1+W_2+W_3)^2]=\frac{100}3+\frac{400}3+300+2(50+75+150)=\frac{3050}3`

giving a variance of

`\mathrm{Var}[W_1+W_2+W_3]=\frac{3050}3-30^2=\frac{350}3`

and so the standard deviation is
`\sqrt{\frac{350}3}\approx\boxed{116.67}`

# # #

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]`

`=\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])`

and since the
`W_i` 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]`

`\mathrm{Var}[W_1+W_2+W_3]=\frac{25}3+\frac{100}3+75=\frac{350}3`

and take the square root to get the standard deviation.