1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
iVinArrow [24]
2 years ago
11

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:
Mathematics
1 answer:
nadya68 [22]2 years ago
8 0

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}\dfrac1{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}\frac{w^2}{10i}\,\mathrm dw=\frac{100i^2}3

so that

\mathrm{Var}[W_i]=\dfrac{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\neq j, and so

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

giving a variance of

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

and so the standard deviation is \sqrt{\dfrac{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]=\dfrac{25}3+\dfrac{100}3+75=\dfrac{350}3

and take the square root to get the standard deviation.

You might be interested in
The square root of 128 times r to the power or three​
mart [117]
<h2>8r square root of 2r</h2>
6 0
3 years ago
(10x2 + 99x – 5) ÷ (x + 10) =
Gennadij [26K]
The answer is 15+99x
______
x+10
5 0
2 years ago
Read 2 more answers
PLZ HELP WILL GIVE BRAINIEST 20 POINTS!!!!!
luda_lava [24]

the answer is y=22.000

3 0
3 years ago
*<br> Simplify the expression:<br> (-6p+7). -4
Art [367]

Answer:

I'm assuming you need to distribute. When simplified the expression is 42p-28

Step-by-step explanation:

7 0
2 years ago
Read 2 more answers
3 soccer balls and 5 more how many are there​
vodomira [7]

Answer:

8

Step-by-step explanation:

3+5=8

6 0
2 years ago
Other questions:
  • A mandatory competency test for high school sophomores has a normal distribution with a mean of 400 and a standard deviation of
    12·1 answer
  • Across a horizontal distance of 25 feet, a roller coaster has a steep drop. The height of the roller coaster at the bottom of th
    13·1 answer
  • Solving Linear Equations SUDOKU<br>help me figure this out please ​
    6·1 answer
  • The manual for your boat engine calls 91 octane gas. The gas stations by your house only sell 87 and 93 octane. If the boat's ta
    10·1 answer
  • Ruby uses 1/4 yard of ribbon to make 2 Bows. Which expression shows the length of ribbon in each bow ?
    14·1 answer
  • A company makes greeting cards and their research shows that that price and demand are related linearly: p=mx +b.They know that
    8·1 answer
  • A gardener has 27 pansies and 36 daises. He plants and an equal number of each type of flower in each row. What is the greatest
    15·1 answer
  • In ∆DEF, DE = 18, EF = 16, and DF = 12. Find m∠F.
    8·1 answer
  • Brainly for the correct answer!<br> Is <img src="https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B0%7D" id="TexFormula1" title="\frac{1}
    11·1 answer
  • The product of the third and the sixth terms of an arithmetic sequence is 406. The
    10·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!