Let Z ~ N(0, 1) and X ~ N(μ σ2) This means that Z is a standard normal random variable with mean 0 and variance 1 while X is a normal random variable with mean μ and variance σ2 (a) Calculate E(Z3) (this is the third moment of Z) b) Calculate E(X) Hint: Do not integrate with the density function of X unless you like messy integration. Instead use the fact that X-eZ + μ and expand the cube inside the expectation.

Answers

Answer 1

a) The third moment of Z is zero. b) E[X] = μ + σ^2μ/3.

(a) To find the third moment of Z, we need to calculate E(Z^3):

Using the formula for the moment generating function of a standard normal distribution:

M(t) = E(e^(tZ)) = exp(t^2/2)

We can differentiate the moment generating function three times to get the third moment:

M''(t) = E(Z^2 e^(tZ)) = (t^2 + 1) exp(t^2/2)

M'''(t) = E(Z^3 e^(tZ)) = (t^3 + 3t) exp(t^2/2)

Therefore, E(Z^3) = M'''(0) = 0 + 3(0) = 0

So, the third moment of Z is zero.

(b) To find E(X), we can use the fact that X = μ + σZ.

Expanding the cube of X - μ in terms of Z, we get:

(X - μ)^3 = (σZ)^3 + 3(σZ)^2 (X - μ) + 3σZ(X - μ)^2 + (X - μ)^3

Taking the expectation of both sides and using linearity of expectation, we get:

E[(X - μ)^3] = E[(σZ)^3] + 3σE[(σZ)^2]E[X - μ] + 3σE[Z](E[X^2] - 2μE[X] + μ^2) + E[(X - μ)^3]

Since Z is a standard normal variable with mean 0 and variance 1, we have:

E[(σZ)^3] = σ^3 E[Z^3] = 0 (from part (a))

E[(σZ)^2] = σ^2 E[Z^2] = σ^2

E[Z] = 0

Also, we know that X is a normal random variable with mean μ and variance σ^2, so:

E[X] = μ

And,

E[X^2] = Var(X) + E[X]^2 = σ^2 + μ^2

Substituting these values into the equation above, we get:

E[(X - μ)^3] = 3σ^2μ + E[(X - μ)^3]

Solving for E[X], we get:

E[X] = μ + σ^2μ/3

Therefore, E[X] = μ + σ^2μ/3.

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Related Questions

Plsssss help me, It says find the area

Answers

Answer:

It should be 136

Step-by-step explanation: I split it up into multiple pieces

Rectangle 1= 20*2=40

Rectangle 2= 14*4=56

Rectangle 3= 20*2=40

40+40+56=136

the area is 136
divide the figure into three rectangles and add all the areas to get the total area

what is the square root of 169​

Answers

Answer: 13

Step-by-step explanation:


the equati...
Solve the equation 3(x + 4.5) = 36.
If you get stuck, use the diagram.
36
4.5
X = ?

Answers

Answer:

x = 7.5

Step-by-step explanation:

Answer:

75

Step-by-step explanation:

if you put 3 in the x it is now 3 but the x it is there with the 3 and do 3 with the 4.5 and then add them to each of them then you get your answer

The table represents some points on the graph of the exponential function that models the radioactive decay of a sample of xenon-135.

Which statement about the graph of this function is true?

(See the picture below)

A) There is an asymptote at y = 70.

B) The y-intercept is located at (0, 70).

C) There is an asymptote at x = 0.

D) The x-intercept is located at (70, 0).

Answers

Answer: b the y-intercept is located at (0,70)

The exponential curve passes through the point (0, 70). Then the correct option is B.

What is an exponent?

Let a be the initial value and x be the power of the exponent function and b be the increasing factor.

The exponent is given as

y = a(b)ˣ

The table addresses a few focuses on the diagram of the outstanding capability that models the radioactive rot of an example of xenon-135.

At x = 4.6, the value of 'y' will be 49.5.

[tex]\rm 49.5 = ab^{4.6} \ \ \ \ \ \ \ ...1\\[/tex]

At x = 9.2, the value of 'y' will be 35.

[tex]\rm 35= ab^{9.2} \ \ \ \ \ \ \ ...2[/tex]

From equations 1 and 2, then we have

[tex]\rm \dfrac{b^{9.2}}{b^{4.6}} = \dfrac{35}{49.5}\\\\b^{4.6} = 0.707\\\\b = 0.927\\[/tex]

Then the value of 'a' will be

[tex]\rm 49.5 = a (0.927)^{4.6}\\\\a = 70[/tex]

Then the equation is given as,

y = 70 · (0.927)ˣ

The graph is given below.

The exponential curve passes through the point (0, 70). Then the correct option is B.

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Use a unit multiplier to convert 0.37 liters to milliliters.

Answers

Answer:

370

Step-by-step explanation:

Multiply the number of liters (L) by 1,000 to find the number of milliliters (mL). There are 1,000 times as many milliliters as there are liters. For instance, say you have 3 liters. Simply multiply 3 liters by 1,000 to get 3,000 milliliters.

Plzz write with formula and explanation plzzzzzzzz answer fast

Answers

Answer:

Rs 19 1/8

Step-by-step explanation:

Fromm the above question, we are told that:

The cost of 2/3 meters of rope = Rs 12 3/4

We are asked to calculate the cost per meter of rope.

Hence:

2/3 meters of rope = Rs 12 3/4

1 meter of rope = Rs x

Cross Multiply

2/3x = 1 × 12 3/4

x = 12 3/4 ÷ 2/3

x = 51/4 ÷ 2/3

x = 51/4 × 3/2

x = 153/8

x = Rs 19 1/8

Therefore, the Cost per meter of rope = Rs 19 1/8

12 parents volunteered to
chaperone the dance. If 15% of all
parents polled volunteered, how
many parents were asked to
volunteer?

Answers

Answer:

80

Step-by-step explanation:

15/100x=12

3/20x=12

x=12*20/3=80

if my answer helped please mark as brainliest.

Answer:

if 15% of all parents polled volunteered, then 80 parents were asked to volunteer.

Step-by-step explanation:

Okay so basically 12 is 15% of a number therefore 12/? = 15%/100?

So we multiply 12 x 100= 1200

Then divide 1200 ÷ 15 to get your answer.

1200 ÷ 15 = 80

Ella is 66 inches tall. Andy is 6 1/2 feet tall. Who is taller? By how much? (12 inches = 1 foot)

Answers

Answer:

Andy is taller by one foot or 12 inches

Step-by-step explanation:

Ella = 66 inches = 5'6"

Andy = 6 1/2 feet = 6'6"

Andy is taller by one foot or 12 inches

Andy is taller by one foot or 12 inches

Given:

Ella is 66 inches and Andy is 6 1/2 feet.

12 inches = 1 feet.

Convert inches in to feet.

12 inches = 1 feet.

1 inches = 1/12 = 0.083

66 inches = (0.083 * 66) = 5.5 feet.

Andy = 6 1/2 feet = 6.5 feet.

6.5 > 5.5 (6.5 is greater than 5.5).

Andy is taller by one feet or 12 inches

Therefore, Andy is taller by one feet.

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someone on this god forsaken planet help me

Answers

Answer: Lol

B (-8,-5)

D (-10,-4)

C (-8,3)

Step-by-step explanation:

If it costs $300 to tile a 420 square foot room, how much will it cost to tile a
room with 1,050 square feet?

Answers

Answer:

750$

Step-by-step explanation:

[tex]Amount= 300*\frac{1050}{420} =750[/tex]

It will cost approximately $750.01 to tile a room with an area of 1,050 square feet.

To find out how much it will cost to tile a room with 1,050 square feet, we can set up a proportion using the given information.

Cost₁: The cost to tile the first room with an area of 420 square feet, which is $300.

Area₁: The area of the first room, which is 420 square feet.

Cost₂: The cost to tile the second room with an unknown area, which we want to find.

Area₂: The area of the second room, which is 1,050 square feet.

We can set up the proportion:

Cost₁ / Area₁ = Cost₂ / Area₂

Substituting the known values:

$300 / 420 sq ft = Cost₂ / 1,050 sq ft

Now we can solve for Cost₂:

Cost₂ = ($300 / 420 sq ft) * 1,050 sq ft

Calculating:

Cost₂ = $0.7143 * 1,050 sq ft

Cost₂ ≈ $750.01

Therefore, it will cost approximately $750.01 to tile a room with an area of 1,050 square feet.

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helppppp pleaaaaaseee

Answers

It is b because you can see how it works and stuff

Help me please answer all thanks

Answers

The coordinates are 3,3

Answer:

the coordinates are (3,3)

Step-by-step explanation:

hope I helped!!

2) Solve the System of Equations using Substitution.
-3x+2y=12
x=2y-8

Answers

Answer:

x=-2. y=3

Step-by-step explanation:

So, it tells us that x=2y-8.

Now substitute x into the equation.

-3(2y-8)+2y=12

- 6y+24+2y=12

-4y+24=12

-4y=-12

y=-12/-4

y=3

Now that we know that y=3 substitute y into the second equation equation.

x=2(3)-8

x=6-8

x=-2

So in the end x=-2 and y=3

Hope this helps

Answer:

y=3, and x= -2

Step-by-step explanation:

-3x= -2y+12

x=2y-8

3x=2y-12

3x=6y-24

2y-12=6y-24

-4y=-12

y=3

x= -2

GUYS PLEASE ANSWER THIS QUESTION BELOW. I AM 1 BRAINLIEST AWAY FROM EXPERT AND NEED SOMEONE ELSE TO ANSWER THIS QUESTION SO I CAN GET BRAINLIEST! YOU DON'T NEED TO ANSWER IT RIGHT JUST TYPE SOMETHING RANDOM!! Thank you!!

https://brainly.com/question/21569168?exp=25-1

Answers

Answer:

Done :)

Step-by-step explanation:

Best of luck!

Oh okie so cute ☺️ cute lol

Hey I'm Chloe Lol, Can You Help Me Thank you :)
Solve The following For x :
3( 3x+9) = 4(4 x+5)

Answers

3x-9=4

Step-by-step explanation:

Answer:

x = 1

Step-by-step explanation:

3(3x+9)=4(4x+5) distribute through multiplication

9x + 27= 16x + 20 subtract 9x and 20 from both sides

7 = 7x divide both sides by 7

x =1

check work

3(3(1)+9) = 4(4(1)+5) distribute the x

3(12) = 4(9)

36 = 36

Congrats I hope this helped!

The population of a city has increased by 38% since it was last measured. If the current population is 89,700, what was the previous population.

Answers

Answer:

65000

Step-by-step explanation:

Given data

Let the previous population be x

the increase was by 38%

The expression below describes the current population

38/100*x+ x=89700

0.38x+x= 89700

1.38x= 89700

x= 89700/1.38

x= 65000

Hence the previous population is 65000

Michael sells 136 newspapers each week. If there are 52 weeks in a year, how many newspapers does Michael sell in a year?

Answers

Answer:

7,072

Step-by-step explanation:

136 times 52 = 7,072

Answer:

7,072 newspaper in a year

Step-by-step explanation:

Multiply: 136*52=7,072

(I would put a picture of my work but I'm lazy right now lol)

Hope this helped!! :)

Stay safe and have a wonderful day/night!!!!!

Brainliest?!?!

Find the solution of the equation. 2(2x + 3) = -4 + 2(2x + 5)

Answers

Answer:

x = 2/3

Step-by-step explanation:

4x + 6 = 10x + 10

4x = 10x - 4

6x - 4

x = 2/3

Have a great day <3

2(2x + 3) = -4 + 2(2x + 5)
Divide both sides by 2
2x + 3 = -2 + (2x + 5)
Add 2 to both sides
2x + 3 + 2 = 2x + 5
2x + 5 = 2x + 5
Subtract 5 from both sides
2x = 2x
Divide both sides by 2
x = x

This means x can equal anything I believe.

what is the sum of 3aa+5aa=

Answers

Answer:

8aa

Step-by-step explanation:

Answer:

8aa is the correct answer of this question

What is the solution to the equation? ​

Answers

X=63


Hope this helps you ☺️

Answer:

C. x = 63

Step-by-step explanation:

X ÷ 3 = 21 is the same as 3 × 21 which is 63.

Evaluate the expression when when c = 3 and d = 33

Answers

Answer:

45

Step-by-step explanation:

define el concepto de renacimiento y el de humanismo​

Answers

Answer:

Renacimiento es el nombre dado en el siglo XIX a un amplio movimiento cultural que se produjo en Europa Occidental durante los siglos XV y XVI. ... El Renacimiento fue fruto de la difusión de las ideas del humanismo, que determinaron una nueva concepción del hombre y del mundo.

Step-by-step explanation:

Please give thanks to all my answers and please mark as brilliant and please follow me

Find the x- and y-intercepts of the graph of x + 5y = 6 state your answer as whole numbers or as improper fractions in simplest form

Answers

Answer:

[tex]y=-\frac{1}{5}x+\frac{6}{5}[/tex]

Step-by-step explanation:

Let's solve for x.

[tex]x+5y=6[/tex]

Step 1: Add -5y to both sides.

[tex]x+5y+-5y=6+-5y[/tex]

[tex]x=-5y+6[/tex]

Answer:

[tex]x=-5y+6[/tex]

Let's solve for y.

[tex]x+5y=6[/tex]

Step 1: Add -x to both sides.

[tex]x+5y+-x=6+-x[/tex]

[tex]5y=-x+6[/tex]

Step 2: Divide both sides by 5.

[tex]\frac{5y}{5}=\frac{-x+6}{5}[/tex]

[tex]y=\frac{-1}{5} x+\frac{6}{5}[/tex]

Answer:

[tex]y=\frac{-1}{5} x+\frac{6}{5}[/tex]

Help PLEASE I forgot how to do this :'(​

Answers

Answer:

C) The sum of the first 2 terms after the fourth term.

PLEASE ANSWER ASAP!

Find (f+g)(x).

f(x) = 1/x

g(x) = x² + 2x - 5

-----------------------

a.) 1/x - x² - 2x b.) 1/x + x² + 2x - 5

c.) x² + 2x -5 - 1/x d.) 1/x + x² - 5​

Answers

Answer:

I think it is b but I could be wrong

NEE HELP!

Solve for x, round your answer to the nearest hundredth: 5lnx + 2 = 17

*need step-by-step solution

Answers

Answer:

e^3

Step-by-step explanation:

5ln(x)+2=17

5ln(x)=17-2

5ln(x)=15

ln(x)=15/5

ln(x)=3

e^ln(x)=e^3

x=e^3

can some one help me please....
by step by step explanation...... ​

Answers

The answer is A) 4.step by step explanations is in the picture.

Quadrilateral ABCD has side lengths of 2, 5, 7, and 12. A
new quadrilateral is created, using a scale factor of 0.25.
Which of the following are side lengths of the new
quadrilateral?
Quick first gets brainliest!!

Answers

Answer:

You didnt put any answer choices but by applying that scale factor, I got these measurements in the order you put them in. 0.5. 1.25. 1.75. 3.

Simplify
(2m-m) - (3m+4n)

Answers

-2m-4n that’s the answer.

i need helpppppppppppppp​

Answers

Answer:

B

Step-by-step explanation:

Factor out 1/4.

1/4 - 3/4 x = (1/4)(1 - 3x)

Answer: B

b because 1/4*1 = 1/4 and 1/4*-3 = -3/4
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