A continuous random variable X has probability density function X. Show how its
moment generating function can be used to determine the variance of this random
variable.

Answers

Answer 1

The moment generating function is defined by

[tex]M_X(t) = \mathbb E[e^{tX}][/tex]

Recall the power series expansion for the exponential function:

[tex]\displaystyle \sum_{n=0}^\infty \frac{x^n}{n!} = 1 + x + \frac{x^2}2 + \frac{x^3}6 + \cdots[/tex]

Then by extension, the MGF could be similarly written as

[tex]\displaystyle M_X(t) = \mathbb E \left[1 + Xt + \frac{(Xt)^2}2 + \frac{(Xt)^3}6 + \cdots\right][/tex]

There's a certain theorem (due to Fubini, in case you're interested in learning more about it) that let's us exchange the order of integration (recall the definition of expectation for continuous random variables) and summation, so that

[tex]\displaystyle M_X(t) = \mathbb E[1] + \mathbb E[Xt] + \mathbb E\left[\frac{X^2t^2}2\right] + \mathbb E\left[\frac{X^3t^3}6\right] + \cdots[/tex]

and by the linearity of expectation,

[tex]\displaystyle M_X(t) = 1 + \mathbb E[X] t + \frac12 \mathbb E\left[X^2\right] t^2 + \frac16 \mathbb E\left[X^3\right] t^3 + \cdots[/tex]

and here we see where the name MGF comes from: the coefficient of the n-th order term in the series expansion "generates" the n-th moment, which is defined as E[Xⁿ].

Now, recall the definition of variance:

[tex]\mathrm{Var}(X) = \mathbb E\left[\left(X - \mathbb E[X]\right)^2\right][/tex]

[tex]\mathrm{Var}(X) = \mathbb E\left[X^2\right] - \mathbb E[X]^2[/tex]

and this is exactly the difference between the second moment and the square of the first moment.

So if you know the MGF, then you essentially get the variance for free with little effort. By differentiating the MGF, we get

[tex]\displaystyle M_X''(t) = \mathbb E[X] + \mathbb E\left[X^2\right] t + \frac12 \mathbb E\left[X^3\right] t^2 + \cdots[/tex]

and setting t = 0 lets us recover the first moment, E[X].

Differentiating again gives

[tex]\displaystyle M_X'(t) = \mathbb E\left[X^2\right] + \mathbb E\left[X^3\right] t + \cdots[/tex]

and setting t = 0 once again recovers the second moment.

Then in terms of the MGF, we have

[tex]\boxed{\mathrm{Var}(X) = M_X''(0) - M_X'(0)^2}[/tex]


Related Questions

HELP
The perimeter of a rectangular pool is 740 yd. If the length is 30 yd less than four times the width, find the length and width.


The length of the rectangular pool is __ yd.

Answers

The length of the rectangle is 12 yd

pls answer this .............

Answers

Answer:

Do the same thing for the other exercises and you should find the results. They are all complimentary angles, that means if you sum the two equations representing the angles of each exercise they will be equal to 180º always

Step-by-step explanation:

3. notice 2x is complimentary of 2x + 80 because 2x is opposed to that blank space that compliments it.... I'm bad at explaining, I'll just show.

2x + 2x + 80 = 180

4x + 80 = 180

4x = 180 - 80

4x = 100

x = 25º

so then:

2x = 2*25

2x = 50º

and:

2x + 80 = 50 + 80

2x + 80 = 130º

QUIEN MONDA ME AYUDA A SOLUCIONA'
f(x)=(3x-4x)(-8x² + 3x)
POR FAVOR

Answers

Answer:

- x(- 8x^2 + 3x)

8x^3 - 3x^2

When a divisor divides the dividend exactly without leaving remainder then the divisor and quotient are
pls answer​

Answers

Answer:

The divisor is called a factor of the dividend.

Step-by-step explanation:

Hope this helps

(Brainliest is appreciated)

Answer:

If a divisor divides the dividend exactly and leaves no remainder, this divisor is also called a factor of that dividend. Therefore, all factors of a number are divisors, but not always all factors have to be factors of a number The factors of 8 are for example 1, 2, 4 and 8, because 8 can be divided exactly by 1, 2, 4, 8.

Step-by-step explanation:

Hope this helps :)

correct?////////////////

Answers

Answer:

No solution

One solution

Many solutions

One solution

Step-by-step explanation:

Let's use y = mx + b

m = coefficient (or slope)

b = constant ( or y-intercept)

If they have different y-intercepts but they're both rising the same amount there would be no solution because they would never cross

If the slope was different but the same y-intercept they would cross on the y-axis, making them have one solution

Equal coefficients and constants is correct, there would be infinite solutions

Different slopes and different y-intercepts would make them cross randomly one, so one solution

Hope this helped! Comment if it does or not :)

Solve for t in the following equation.
10 + 4–2 = 2–2( + 4)
A. =−1 B. =−35
C. =35

Answers

Answer:

the answer would be 35...

Step-by-step explanation:

Dont know how to explain it on here

What type of correlation does the data set display?
х
y
2
13
5 11
7 10
9
8
11
2
A. Negative
O B. No correlation
C. Positive

Answers

Answer:

B-NO CORRELATION.....

Pls help I am desperate

Answers

follow the steps shown on the photo

9514 1404 393

Answer:

  7/5x -1/3

Step-by-step explanation:

We assume you want to simplify this expression.

  [tex]\dfrac{-3}{5}x-\dfrac{1}{4}\cdot\left(-8x+\dfrac{5}{6}\right)-\dfrac{1}{8}\\\\=-\dfrac{3}{5}x+\left(-\dfrac{1}{4}\right)\cdot(-8x)+\left(-\dfrac{1}{4}\right)\cdot\left(\dfrac{5}{6}\right)-\dfrac{1}{8}\\\\=\left(-\dfrac{3}{5}+2\right)x-\left(\dfrac{5}{24}+\dfrac{1}{8}\right)\\\\=\dfrac{7}{5}x-\dfrac{5+3}{24}=\boxed{\dfrac{7}{5}x-\dfrac{1}{3}}[/tex]

_____

Additional comment

2 = 10/5; 1/8 = 3/24; 8/24 = 1/3. The x-terms get combined and the constants get combined. It is generally helpful to eliminate parentheses first using the distributive property.

100 points, Please help!

Answers

Answer:

Look if you say 100 points. Give a hundred points. If you want an answer be honest~ and say fifty points

Step-by-step explanation:

1=17

2=97

Answered:

1 = 17

2 = 97

What is the value of x? Enter your answer in the box. X=​

Answers

Check the picture below.

Answer:

x=50

Step-by-step explanation:

1. equal the equations to each other: 2(x+10)=3x-30

2. distribute the 2 to x+10: 2x+20=3x-30

3. subtract 2x from both sides, and add 30 to both sides: x=50

Bài 4: 350 Cho tam giác ABC nhọn, hai đường cao BD, CE cắt nhau tại H. M là trung điểm của BC. K đối xứng với H qua M. a) Chứng minh tứ giác BHCKlà hình bình hành b) Chứng minh BK LAB và CK AC. c) Gọi 0 là trung điểm của AK. Chứng minh OM IBC d) Gọi N là điểm đối xứng với H qua BC. Chứng minh BNKC là hình thang cần​

Answers

no sabo la respuesta jajaja

What is the value of x? Midsegments of triangles

Answers

Hi how u doin ?

[tex] \frac{16}{x + 4} = \frac{1}{2} \\ [/tex]

inverse both sides

[tex] \frac{x + 4}{16} = \frac{2}{1} \\ [/tex]

Multiply both sides by 16

[tex]16 \times \frac{x + 4}{16} = 16 \times 2 \\ [/tex]

[tex]x + 4 = 32[/tex]

Subtract both sides 4

[tex]x + 4 - 4 = 32 - 4[/tex]

[tex]x = 28[/tex]

There u go..... Have a great day ❤

this is just and addition to the great posting below, just supplementing.

keeping in mind that the midsegment of a triangle is half the length of the parallel base/side, Check the picture below.

Danielle's text messaging plan costs $8 for the first 550 messages and 15¢ for each additional text message. If she owes $8.60 for text messaging in the month of October, how many text messages did she send that month?

Answers

Answer:

Danielle sent 554 text messages that month.

Step-by-step explanation:

In Oct she owed: $8 + 60¢

Because 15¢ = 1 additional message => 60¢ = 4 additional messages

So Danielle sent 550 messages (=$8) and 4 additional messages (=  60¢)

=> She sent: 550 messages + 4 additional messages = 554 messages

Hope this helps!

Helppppppppppppppppppppppppppppp

Answers

Answer:

Step-by-step explanation:

just find a new photo

Bradley is 27 years older than Madison seven years go, the sum of their ages was 59?

Answers

Answer:

how old is Madison right then I can't really help u solve it with that little info

Help me poeaseeeeeeeeeeee

Answers

Answer:

C. Hope It's Help :)))))

Answer:

4.5

Step-by-step explanation:

∛100 = ?

100 = 2 x 2 x 5 x 5

∛100

= ∛ ( 2 x 2 x 5 x 5 )

= 4.642

4.642 ~ 4.5

The approximation 4.642 can also be written as 4.5

Therefore,

∛100 = 4.5

In the 7th grade, 30% of students wear a medium size shirt. If 93 students wear a medium, how many total students are there in the 7th grade?

Answers

Answer:

310

Step-by-step explanation:

Answer:

310

Step-by-step explanation:

is/of = % / 100

[tex]93/x = 30/100[/tex]

93 * 100=9300

9300/30 = 310

Kaitlyn purchased a new car 39 months ago for $18791. She has been paying $197.43 per month. How much does Kaitlyn still owe for her car?

Answers

Answer:

$11091.23

Step-by-step explanation:

Reword the situation:

She purchased a car for 18791 dollars 39 months ago.

As the 39 months passed she has been paying 197.43 a month.

How much does she still owe for her car?

First find out how much she's paid so far.

For 39 months she has been paying 197.43 a month

So, so far she's payed a total of 197.43 * 39 = 7699.77 dollars

Now subtract amount she's paid so far from total price of car.

18791 - 7699.77 = 11091.23

She still owes 11091.23

shola earns #750 every month. if she spends 4/5 of her salary and saves the rest, how much does she save every month?​

Answers

Answer:

250

Step-by-step explanation:

Amount she earns = 750

amount she spend = ⅘ of 750 = ⅘×750= 600

amount she saves = 750-600

Shola saves $150 every month.

What is multiplication?

Multiplication is a mathematical arithmetic operation. It is also a process of adding the same types of expression for some number of times.

Example - 2 × 3 means 2 is multiplied by 3, or 2 is added 3 times.

Given:

Shola earns $750 every month.

She spends 4/5 of her salary.

That means,

4/5 of $750.

And 4/5 of $750 = 4/5 x 750

                            = 4 x 150

                            = 600

She spends $650 per month.

Saved amount,

= $750 - $600

= $150

Therefore, $150 is her monthly savings.

To learn more about the multiplication;

https://brainly.com/question/19634536

#SPJ2

Add.



5/6 + 2 3/4



Enter the sum in the box as a mixed number in the simplest form.

Answers

Answer:

7 1/

6

≅ 7.1666667

Step-by-step explanation:

Hope this helps :)

i think the answer is 4 7/12

First round 0.69 to the nearest tenth.

Answers

Answer:

0.69 rounded to the nearest tenth is 0.7

0.7 is the answer because you round the 69 round to 70.


5. Hamza chopped up
pineapple and gave į to his
mum. He also ate half himself.
How much was left to give to
his dad?

Answers

Answer:

depends how much he chopped up

Step-by-step explanation:

Answer:

Nothing

Step-by-step explanation:

"5. Hamza chopped up

pineapple and gave ½ to his

mum. He also ate half himself.

How much was left to give to

his dad?"

1 - 1/2 - 1/2 = 2/2 - 1/2 - 1/2 = 1/2 - 1/2 = 0

Answer: There was nothing left for him.

Use point-slope form to write the equation of a line that passes through the point (19,-17) with slop -5/3

Answers

Answer:

  y +17 = -5/3(x -19)

Step-by-step explanation:

The point-slope form of the equation for a line is ...

  y -k = m(x -h) . . . . . . . line with slope m through point (h, k)

You want a line with slope -5/3 through point (19, -17). Its equation is ...

  y -(-17) = -5/3(x -19) . . . . . substitute the given values

  y +17 = -5/3(x -19) . . . . . simplify

cost of 1 almirah is=$2345
what is the cost of 12 such almirah?​

Answers

Answer:

28140

Step-by-step explanation:

We can applyUnitary method to solve this :-

cost of 1 almirah = $ 2345

therefore,

cost of 12 such almirah = 2345 x 12.

= $28140...ans

Hope it helps you!!

A ladybug crawls around the outside of a rectangle that measures 7.5 cm by 8.5 cm. The ladybug stays exactly 3.9 cm from the rectangle at all times to form a larger figure. What is the area of the new rectangle bound by the path of the bug?

Answers

Answer:

185.64 [tex]cm^{2}[/tex]

Step-by-step explanation:

Width =7.5+3.9*2 = 15.3 cm

Length = 8.5+3.9*2=16.3 cm

Big Area = 15.3*16.3 = 249.39 [tex]cm^{2}[/tex]

Small Area = 7.5*8.5 = 63.75 1 [tex]cm^{2}[/tex]

area bounded by the path of the bug = 249.39-63.75 = 185.64 [tex]cm^{2}[/tex]

(I am giving branlest) (plz help only if you understand )



Solve one seventh plus nine fourteenths equals blank


A ten fourteenths

B eleven fourteenths

C ten sevenths

D eleven sevenths

Answers

Answer:

its 11/14 so its B hope this helps

thanks ill mark brainliest​

Answers

If I'm watching Gossip Girl, then it is Monday.

Hope this helps you!

Answer:

if i am watching Gossip   Girl then it is monday

Step-by-step explanation:

that is the common sense that everyone should know

hope this helps everyone and have a great summer!

A company makes a profit of $y (in thousand dollars) when it produces x computers,
where y is given by the formula y = a(x - 100)(x - 200) for x 20 If 120
computers are produced, the profit will be $3,200,000.
a) Find the value of a.

b) What is the maximum profit the company can make? At this profit, how many
computers should be produced?

c) If the company targets to make at least $4,800,000, what is the range of the
number of computers to be produced?


Answers

Using quadratic function concepts, it is found that:

a) The value of a is -2000.

b) The maximum profit the company can make is of $5,000,000, when 150 computers should be produced.

c) Between 140 and 160 computers need to be produced.

The profit is modeled by:

[tex]y = a(x - 100)(x - 200)[/tex]

Item a:

If 120  computers are produced, the profit will be $3,200,000, hence when [tex]x = 120, y = 3200000[/tex], and this is used to find a.

[tex]y = a(x - 100)(x - 200)[/tex]

[tex]3200000 = a(120 - 100)(120 - 200)[/tex]

[tex]-1600a = 3200000[/tex]

[tex]a = -\frac{3200000}{1600}[/tex]

[tex]a = -2000[/tex]

The value of a is -2000.

Item b:

We first place the quadratic function into standard form, thus:

[tex]y = -2000(x - 100)(x - 200)[/tex]

[tex]y = -2000(x^2 - 300x + 20000)[/tex]

[tex]y = -2000x^2 + 600000x - 40000000[/tex]

Which has coefficients [tex]a = -2000, b = 600000, c = -40000000[/tex].

Then, we have to find the vertex:

[tex]x_V = -\frac{b}{2a} = -\frac{600000}{2(-2000)} = 150[/tex]

[tex]\Delta = b^2 - 4ac = (600000)^2 - 4(-2000)(-40000000) = 40000000000 [/tex]

[tex]y_V = -\frac{\Delta}{4a} = -\frac{40000000000 }{4(-2000)} = 5000000[/tex]

The maximum profit the company can make is of $5,000,000, when 150 computers should be produced.

Item c:

We are working with a concave down parabola, hence the range is between the roots of:

[tex]y = -200x^2 + 600000x - 40000000[/tex]

[tex]4800000 = -200x^2 + 600000x - 40000000[/tex]

[tex]-200x^2 + 600000x - 44800000 = 0[/tex]

The coefficients are [tex]a = -200, b = 600000, c = -44800000[/tex].

Then:

[tex]\Delta = b^2 - 4ac = (600000)^2 - 4(-2000)(-44800000) = 1600000000[/tex]

[tex]x_1 = \frac{-b - \sqrt{Delta}}{2a} = \frac{-600000 - \sqrt{1600000000}}{2(-2000)} = 160[/tex]

[tex]x_2 = \frac{-b + \sqrt{Delta}}{2a} = \frac{-600000 + \sqrt{1600000000}}{2(-2000)} = 140[/tex]

Between 140 and 160 computers need to be produced.

A similar problem is given at https://brainly.com/question/24705734

4 lbs of tomato’s cost $14 complete the double number line so that it shows how much it will cost for up to 8 lbs of tomato’s

Answers

Answer:

the answer is 28$ because since the lbs get doubled so ypu also have to double the cost

Joy has ₱500.00 to spend for food. She spent ₱98.75 on Monday, ₱73.75 on Tuesday and ₱50.25 on Wednesday. How much money was left? with solution pls​

Answers

Answer:

₱277.25

Step-by-step explanation:

₱500.00 - ₱98.75 - ₱73.75 - ₱50.25 = ₱277.25

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