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
dedylja [7]
2 years ago
13

The lifetime X (in hundreds of hours) of a certain type of vacuum tube has a Weibull distribution with parameters α = 2 and β =

3.† Compute the following. (Round your answers to three decimal places.) (a) E(X) and V(X) E(X) = 2.659 V(X) = 1.931 (b) P(X ≤ 3) .6321 (c) P(1 ≤ X ≤ 3) .5269
Mathematics
1 answer:
stich3 [128]2 years ago
7 0

I'm assuming \alpha is the shape parameter and \beta is the scale parameter. Then the PDF is

f_X(x)=\begin{cases}\dfrac29xe^{-x^2/9}&\text{for }x\ge0\\\\0&\text{otherwise}\end{cases}

a. The expectation is

E[X]=\displaystyle\int_{-\infty}^\infty xf_X(x)\,\mathrm dx=\frac29\int_0^\infty x^2e^{-x^2/9}\,\mathrm dx

To compute this integral, recall the definition of the Gamma function,

\Gamma(x)=\displaystyle\int_0^\infty t^{x-1}e^{-t}\,\mathrm dt

For this particular integral, first integrate by parts, taking

u=x\implies\mathrm du=\mathrm dx

\mathrm dv=xe^{-x^2/9}\,\mathrm dx\implies v=-\dfrac92e^{-x^2/9}

E[X]=\displaystyle-xe^{-x^2/9}\bigg|_0^\infty+\int_0^\infty e^{-x^2/9}\,\mathrm x

E[X]=\displaystyle\int_0^\infty e^{-x^2/9}\,\mathrm dx

Substitute x=3y^{1/2}, so that \mathrm dx=\dfrac32y^{-1/2}\,\mathrm dy:

E[X]=\displaystyle\frac32\int_0^\infty y^{-1/2}e^{-y}\,\mathrm dy

\boxed{E[X]=\dfrac32\Gamma\left(\dfrac12\right)=\dfrac{3\sqrt\pi}2\approx2.659}

The variance is

\mathrm{Var}[X]=E[(X-E[X])^2]=E[X^2-2XE[X]+E[X]^2]=E[X^2]-E[X]^2

The second moment is

E[X^2]=\displaystyle\int_{-\infty}^\infty x^2f_X(x)\,\mathrm dx=\frac29\int_0^\infty x^3e^{-x^2/9}\,\mathrm dx

Integrate by parts, taking

u=x^2\implies\mathrm du=2x\,\mathrm dx

\mathrm dv=xe^{-x^2/9}\,\mathrm dx\implies v=-\dfrac92e^{-x^2/9}

E[X^2]=\displaystyle-x^2e^{-x^2/9}\bigg|_0^\infty+2\int_0^\infty xe^{-x^2/9}\,\mathrm dx

E[X^2]=\displaystyle2\int_0^\infty xe^{-x^2/9}\,\mathrm dx

Substitute x=3y^{1/2} again to get

E[X^2]=\displaystyle9\int_0^\infty e^{-y}\,\mathrm dy=9

Then the variance is

\mathrm{Var}[X]=9-E[X]^2

\boxed{\mathrm{Var}[X]=9-\dfrac94\pi\approx1.931}

b. The probability that X\le3 is

P(X\le 3)=\displaystyle\int_{-\infty}^3f_X(x)\,\mathrm dx=\frac29\int_0^3xe^{-x^2/9}\,\mathrm dx

which can be handled with the same substitution used in part (a). We get

\boxed{P(X\le 3)=\dfrac{e-1}e\approx0.632}

c. Same procedure as in (b). We have

P(1\le X\le3)=P(X\le3)-P(X\le1)

and

P(X\le1)=\displaystyle\int_{-\infty}^1f_X(x)\,\mathrm dx=\frac29\int_0^1xe^{-x^2/9}\,\mathrm dx=\frac{e^{1/9}-1}{e^{1/9}}

Then

\boxed{P(1\le X\le3)=\dfrac{e^{8/9}-1}e\approx0.527}

You might be interested in
A circle has a circumference of 125.6 centimeters. what is the diameter of the circle to the nearest centimeter?what is the radi
natali 33 [55]
You divide 125.6 by 3.14 which = 68.66
That's the diameter. so the nearest cm is 69cm
The radius is half of the diameter so to the nearest cm it is 34cm
5 0
3 years ago
4x-y-32<br> if x=-2 y=3 and z= -1
Radda [10]
The answer is 6 jus do it
6 0
3 years ago
If the Cost price of the an article is greater than the selling price, we have a ____?​
Olegator [25]

Answer:

loss

Step-by-step explanation:

Selling an item for less than was paid gives a loss

Selling an item for more than was paid gives a profit

4 0
3 years ago
Solve for x<br> -30=5(x+1)
mariarad [96]

Answer:

-5=x

Step-by-step explanation:

-30=5x+5

-25=5x

-5=x

5 0
3 years ago
Read 2 more answers
PLEASE HELP ASAP!! Will give brainliest!
Degger [83]
Hi there! 9.125 * 10^-3 is definitely less than 9.125 * 10^2, because when it comes to negative powers of 10, the numbers are decimals. 10^-3 equals 0.001 or one thousandths in word form. 10^2 is 100. When solving for the answers. 0.009125 is less than 912.5. When 912.5 is divided by 0.009125 912.5 is 100^5 or 100,000 times greater than 0.009125. The answer is that 9.125 * 10^-3 is 100,000 (100^5) times less than 9.125 * 10^2.
7 0
3 years ago
Read 2 more answers
Other questions:
  • Q2) The isoscsles right triangle has 2 of it’s sides 10,10 it’s area is
    13·1 answer
  • Lance wants to find the total length of 3 boards.He uses the expression 3 1/2+( 2+4 1/2 how can lance rewrite the expression usi
    9·2 answers
  • What is thei value of their given digits the 4 in 440
    5·1 answer
  • Graph 3x-2y=16 and x-2y=4
    14·1 answer
  • GEOMETRY! Show your work, please.
    14·1 answer
  • Can anyone answer these questions? Look at picture
    5·1 answer
  • Each day, Gisselle bikes to school in the morning and bikes home in the afternoon. If she
    7·2 answers
  • 3c + 2 = -22<br> whats c?
    7·2 answers
  • PLEASE HELP !! ILL GIVE BRAINLIEST !! 100 POINTS
    10·2 answers
  • Tim and sue share a small pizza. tim eats 2/3 o the pizza
    14·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!