The random variable X has the following probability density function: fX(x) = ( xe−x , if x > 0 0, otherwise. (a) Find the moment generating function of X. (b) Using the moment generating function of X, find E[Xn] for all positive integers n. Your final answer should be an expression that depends only on n.

Answers

Answer 1

Answer:

Follows are the solution to the given points:

Step-by-step explanation:

Given value:

[tex]\to f_X (x) \ \ xe^{-x} \ , \ x>0[/tex]

For point a:

Moment generating function of X=?

Using formula:

[tex]\to M(t) =E(e^{tx})= \int^{\infty}_{-\infty} \ e^{tx} f(x) \ dx[/tex]

[tex]M(t) = \int^{\infty}_{-\infty} \ e^{tx}xe^{-x} \ dx = \int^{\infty}_{0} \ xe^{(t-1)x} \ dx[/tex]

integrating the values by parts:

[tex]u = x \\\\dv = e^{(t-1)x}\\\\dx =dx \\\\v= \frac{e^{(t-1)x}}{t-1}\\\\M(t) =[\frac{e^{(t-1)x}}{t-1}]^{-\infty}_{0} -\int^{\infty}_{0} \frac{e^{(t-1)x}}{t-1} \ dx\\\\[/tex]  

        [tex]= \frac{1}{t-1} (0) - [\frac{e^{(t-1)x}}{(t-1)^2}]^{\infty}_{0}\\\\=\frac{1}{(t-1)^2}(0-1)\\\\=\frac{1}{(t-1)^2}\\\\[/tex]

Therefore, the moment value generating by the function is [tex]=\frac{1}{(t-1)^2}[/tex]

In point b:

[tex]E(X^n)=?[/tex]

Using formula: [tex]E(X^n)= M^{n}_{X}(0)[/tex]

form point (a):

[tex]\to M_{X}(t)=\frac{1}{(t-1)^2}[/tex]

Differentiating the value with respect of t

[tex]M'_{X}(t)=\frac{-2}{(t-1)^3}[/tex]

when [tex]t=0[/tex]

[tex]M'_{X}(0)=\frac{-2}{(0-1)^3}= \frac{-2}{(-1)^3}= \frac{-2}{-1}=2\\\\M''_{X}(t)=\frac{(-2)(-3)}{(t-1)^4}\\\\M''_{X}(0)=\frac{(-2)(-3)}{(0-1)^4}= \frac{6}{(-1)^4}= \frac{6}{1}=6\\\\M''_{X}(t)=\frac{(-2)(-3)}{(t-1)^4}\\\\M''_{X}(0)=\frac{(-2)(-3)}{(0-1)^4}= \frac{6}{(-1)^4}= \frac{6}{1}=6\\\\M'''_{X}(t)=\frac{(-2)(-3)(-4)}{(t-1)^5}\\\\M''_{X}(0)=\frac{(-2)(-3)(-4)}{(0-1)^5}= \frac{24}{(-1)^5}= \frac{24}{-1}=-24\\\\\therefore \\\\M^{K}_{X} (t)=\frac{(-2)(-3)(-4).....(k+1)}{(t-1)^{k+2}}\\\\[/tex]

[tex]M^{K}_{X} (0)=\frac{(-2)(-3)(-4).....(k+1)}{(-1)^{k+2}}\\\\\therefore\\\\E(X^n) = \frac{(-2)(-3)(-4).....(n+1)}{(-1)^{n+2}}\\\\[/tex]


Related Questions

Use the table to answer the question.
x 0 13 4 6
y 0 1 9 16 36
Which equation represents the relationship between x and y in the table?

Answers

Answer:

y = x ÷ x

Step-by-step explanation:

x = 13

y = 13 ÷ 13

y = 1

so y = x ÷ x

A number x decreased by 6 is at most 11.​

Answers

Answer:

17?

Step-by-step explanation:

Triangle A is transformed by the
combined transformation of a rotation
of 180° about the point (-1,0) followed
by a translation with vector
-3
2)
One point on triangle A is invariant
under the combined transformation.
What are the coordinates of this point?

Answers

Answer:

( -2.5 , 1 )

Step-by-step explanation:

(-2.5 ,1 ) hope this helps u

18. Which of the following correctly shows 2,354,769 rounded to the nearest ten thousand?​

Answers

as uve not shown no options it shud be round bout 2350000

Step-by-step explanation:

Answer:

2,350,000

Step-by-step explanation:

this is the answer I believe

Allison earns a 10% commission on her weekly sales. Each week, she puts of her commission in a savings account and uses the amount remaining on bills. If Allison's sales for one week were $2,300.00. how much money does she use for bills?​

Answers

Answer:

$2,070 is how much she will use for bills.

help teacher graph only​

Answers

9514 1404 393

Answer:

  see attached

Step-by-step explanation:

A graphing calculator or spreadsheet program can plot these points for you. It's probably no more work to actually plot the points yourself on the given graphs than it is to type them into a program.

As you know, the x-coordinate is the horizontal distance from the y-axis. Positive is to the right.

The y-coordinate is the vertical distance from the x-axis. Positive is up.

You look up at a 62 m tall building and find that the angle of elevation from the ground to the top of the building is 51 degrees. How far away from the building's base are you?

Answers

Answer:

Step-by-step explanation:

horizontal distance = 62/tan(51°) ≅ 50.2 m

The distance between the building base to a standing point is 50.20 m if the angle of elevation from the ground to the top of the building is 51 degrees.

What is the trigonometric ratio?

The trigonometric ratio is defined as the ratio of the pair of a right-angled triangle.

Trigonometry is a branch of mathematics that deals with the relationship between the sides and angles of a right-angle triangle.

It is given that:

You look up at a 62 m tall building and find that the angle of elevation from the ground to the top of the building is 51 degrees.

The angle = 51 degrees.

The height of the building = 62 m

Applying the tan ratio

tan51 = height of the building/distance between the building base to a standing point

tan51 = 62/x

x = 50.20 m

Thus, the distance between the building base to a standing point is 50.20 m if the angle of elevation from the ground to the top of the building is 51 degrees.

Learn more about trigonometry here:

brainly.com/question/26719838

#SPJ6

simplify the first expression 5(m+1)-1=

Answers

Answer:

-0.8/ -4/5

Step-by-step explanation:

5(m+1) -1 = 0

5m +5 -1 = 0

5m +4 =0

5m= -4

m= -4/5

The diameter of a circle is 4 cm. Find its area in terms of \piπ.

Answers

Answer:

Area of circle = 4π.

Step-by-step explanation:

Given the following data;

Diameter = 4cm

Pie = π.

To find the area of the circle;

Radius = diameter/2

Radius = 4/2

Radius = 2cm

Area of a circle = πr²

Substituting into the equation, we have;

Area of circle = π*2²

Area of circle = 4π.

Can someone help me please ?

Answers

Step-by-step explanation:

For A

The name of two similar triangles is ABC and CED.

For b

If two pairs of corresponding angles in a pair of triangles are congruent, then the triangles are similar. We know this because if two angle pairs are the same, then the third pair must also be equal. When the three angle pairs are all equal, the three pairs of sides must also be in proportion.

Here

< BAC = < CED ( given)

<ACB = < ECD ( vertically opposite angles)

Need help pls!!!!!!!!!!?

Answers

Answer:

2, 4, & 6 = 84 degrees

1, 3, 5 , & 7 = 96 degrees

Step-by-step explanation:

Solve for r in the expression 3 (2 - 5) = 12.​

Answers

Answer:

The answer is x = 9

Step-by-step explanation:

[tex]3(x-5)=12[/tex]

use the distributive property on the left side

[tex]3x-15=12[/tex]

add 15 to both sides

[tex]3x=27[/tex]

divide both sides by 3

[tex]x = 9[/tex]

Answer:

The correct answer will be x=9

Step-by-step explanation:

Because I said so

7. If it takes 2 gardeners 6 hours to landscape a garden, how long would it take 3 gardeners?

12

8. Question 7 is an example of...

Direct proportion
Inverse proportion​

Answers

the answer is inverse proportion

a woman spent 3/5 of her weekly salary on rent and 25% of her weekly salary on food. Her food cost $2500. How much did she spend on rent

Answers

Answer:

$6,000

Step-by-step explanation:

If her food costed $2500 and was 25%, then she had $10,000 to begin with. 3/5 of $10,000 is $6,000, so her rent costs $6,000.

Answer:

$6,000 on rent

Step-by-step explanation:

Find her weekly salary by dividing 2500 by 0.25

2500/0.25

= 10,000

Now, find how much she spent on rent by multiplying 10,000 by 3/5

10,000(3/5)

= 6,000

So, she spent $6,000 on rent

Find the probability of the indicated event if P(E) = 0.25 and P(F) = 0.40.
Find P(E or F) if P(E and F) = 0.05.
P(E or F) = (Simplify your answer.)

Answers

Answer:

[tex]P(E\ or\ F) = 0.60[/tex]

Step-by-step explanation:

Given

[tex]P(E) = 0.25[/tex]

[tex]P(F) = 0.40[/tex]

[tex]P(E\ and\ F) = 0.05[/tex]

Required

Determine P(E or F)

This is calculated using:

[tex]P(E\ or\ F) = P(E) + P(F) - P(E\ and\ F)[/tex]

[tex]P(E\ or\ F) = 0.25 + 0.40 - 0.05[/tex]

[tex]P(E\ or\ F) = 0.60[/tex]

Bentley invested $430 in an account paying an interest rate of 2.5% compounded continuously. Assuming no deposits or withdrawals are made, how long would it take, to the nearest year, for the value of the account to reach $510?

Answers

Answer:

its 11

Step-by-step explanation:

Answer:

7

Step-by-step explanation:

Store A sells 12 juice bottles for $4 and store B sells 18 juice bottles for $6. Are these rates equivalent? Explain your reasoning.

Answers

Answer:

these rates are equivalent.

12bottles/$4= 18bottles/$6= 3bottles/$1

Step-by-step explanation:

Evaluate the expression (14 − 4)2 ÷ 10.

The value of the expression is value of

Answers

Answer:

2

Step-by-step explanation:

PEMDAS says to do the equations in parentheses first

(14-4) = 10

PEMDAS then says to do multiplication or division, whichever comes first in the equation, which is multiplication in this case

(10)2 = 20

20 ÷ 10 = 2

Convert 18 inches to feet using the table.
inches
conversion
feet

14 inches convert to feet

Answers

Answer:

18inches = 1.5 ft.

14inches= 1.16667 ft.

Step-by-step explanation:

1inch = 0.0833 ft.

Answer:

18 inches = 1.5ft

14 inches = 1.16ft

Step-by-step explanation:

First, you need to know how many inches are in a foot. Then you take what you have and divide it by that number. So it would be 18/12 or 14/12 and that would give you your answer

The width of a rectangle is 2 less than twice its length. If the area of the rectangle is 110
cm
2
, what is the length of the diagonal?

Answers

Answer:

Use the formula of areal*b*h

Find the surface area of the square pyramid shown below.

Answers

Answer:

12

Step-by-step explanation:

6+6=12

3x4=12

A number, x, rounded to 1 decimal place is 3.7
Write down the error interval for x.

Answers

Answer: (3.65, 3.74)

Step-by-step explanation:

Now, when we round a number to a given decimal place, we need to look at the decimal to the left of the decimal place where we are rounding.

if it is 5 or more, we round up

if it is smaller than 5, we round down.

Then for example, if we want to round:

4.534

to the first decimal place, we need to look at the second decimal place.

we can see that it is a 2.

Then we round down, and get:

4.5

Now we know that, when rounding to the first decimal place, we have:

x = 3.7

Then the maximum value that x could have is:

x = 3.74 (because here we round down to 3.7)

and the minimum value that x could have is:

x = 3.65 (because here we round up to 3.7)

Then the error interval for x is:

(3.65, 3.74)

HELPP PLZZZZZZZZZZ tand thanks

Answers

8 ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎

What is the measure of the missing angle?

Answers

Answer:

115

Step-by-step explanation:

To find the missing angle all you have to do is add the 2 nonadjacent interior angles which are 75 and 40 to get the exterior angle which is 115.

Hope this helps!:)

75+40+x=180

X= 65

65 + x= 180

Answer is 115°

Bob wants to buy a TV that costs $100. He plans to save $5 per week. How many weeks w will it take Bob to save $100?

Answers

Answer:

he will have to save 5 dollars for 20 weeks

Step-by-step explanation:

5 x 20 = 100

So if the tv cost 100 dollars if he needs 100 you can do 100/5 and that will get you 20 weeks

Liliana and Marcus are playing a word game. Liliana spells a word incorrectly and loses 18 points. Then Marcus
spells a word correctly and has the opposite score. Which is the correct way to represent the opposite of
Liliana's score is equal to Marcus's score"?
0 -18-18
O-(-18) --18
-(18) --18
-(-18) - 18

Answers

Answer:

The answer is B or -(-18) = 18

Step-by-step explanation:

on Edg 2021

Answer:

D

Step-by-step explanation:

Cuz I did it on a test and I got it right.

2.
What type of a quadrilateral is defined by the vertices (-5, 7), (2,6), (5, -3), (-4, 0)?

a Kite
b Rectangle
C Rhombus
d Parallelogram

Answers

the answer is a kite

Note: Enter your answer and show all the steps that you use to solve this problem in the space provided. Calculate the area of the trapezoid, which is not drawn to scale.

Answers

Answer:

38 in

Step-by-step explanation:

Ok, so the bottom part is a rectangle and the top part is a triangle. We’re gonna split those into two different shapes.

8 * 4 = 32, this is the area of the bottom rectangle.

To get the side length of the top triangle, we have to subtract 8 from 11.

11 - 8 = 3

(3 * 4)/2 = 6, this is the area of the top triangle.

Add the two areas, 32 + 6 = 38

And that’s it, maybe, I think.

I DONT KNOW WHAT IM DOING, PLEASE HELP

Answers

Answer:

you should check out Microsoft Math! It's a free app :)

Please show the work.

1. A wheelchair ramp is 4.2 m long. I rises 0.7 . What is its angle of inclination to the nearest degree?

2. A person flying a kite has released 176 m of string and attached the end to the ground. The string makes an angle of 27 degree with the ground. How high is the kite? How far away is the kite horizontally? Answer to the nearest meter.

Answers

Answer:

see explanation

Step-by-step explanation:

(1)

The angle of inclination is the angle between the horizontal and the ramp.

Using the sine ratio in the right triangle formed by the ground and the ramp

let x be the angle of inclination, then

sinx = [tex]\frac{opposite}{hypotenuse}[/tex] = [tex]\frac{0.7}{4.2}[/tex] , then

x = [tex]sin^{-1}[/tex] ([tex]\frac{0.7}{4.2}[/tex] ) ≈ 10° ( to the nearest degree )

(2)

Again there is a right triangle between the ground and the vertical height of the kite, with hypotenuse being length of string.

let y be the height of the kite and x the horizontal distance, then

sin27° = [tex]\frac{opposite}{hypotenuse}[/tex] = [tex]\frac{y}{176}[/tex] ( multiply both sides by 176 )

176 × sin27° = y , then

y ≈ 80m ( to the nearest metre ) ← height of kite

and

cos27° = [tex]\frac{adjacent}{hypotenuse}[/tex] = [tex]\frac{x}{176}[/tex] ( multiply both sides by 176 )

176 × cos27° = x , then

x ≈ 157m ( to the nearest metre ) ← horizontal distance

Step-by-step explanation:

[tex]1)

The angle of inclination is the angle between the horizontal and the ramp.

Using the sine ratio in the right triangle formed by the ground and the ramp

let x be the angle of inclination, then

sinx = \frac{opposite}{hypotenuse}hypotenuseopposite = \frac{0.7}{4.2}4.20.7 , then

x = sin^{-1}sin−1 (\frac{0.7}{4.2}4.20.7 ) ≈ 10° ( to the nearest degree )

(2)

Again there is a right triangle between the ground and the vertical height of the kite, with hypotenuse being length of string.

let y be the height of the kite and x the horizontal distance, then

sin27° = \frac{opposite}{hypotenuse}hypotenuseopposite = \frac{y}{176}176y ( multiply both sides by 176 )

176 × sin27° = y , then

y ≈ 80m ( to the nearest metre ) ← height of kite

and

cos27° = \frac{adjacent}{hypotenuse}hypotenuseadjacent = \frac{x}{176}176x ( multiply both sides by 176 )

176 × cos27° = x , then

x ≈ 157m ( to the nearest metre ) ← horizontal distance

[/tex]

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