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
vodomira [7]
3 years ago
5

Square of a standard normal: Warmup 1.0 point possible (graded, results hidden) What is the mean ????[????2] and variance ??????

??????[????2] of the random variable ????2? ????[????2]= unanswered ????????????[????2] unanswered
Mathematics
1 answer:
LenaWriter [7]3 years ago
7 0

Answer:

E[X^2]= \frac{2!}{2^1 1!}= 1

Var(X^2)= 3-(1)^2 =2

Step-by-step explanation:

For this case we can use the moment generating function for the normal model given by:

\phi(t) = E[e^{tX}]

And this function is very useful when the distribution analyzed have exponentials and we can write the generating moment function can be write like this:

\phi(t) = C \int_{R} e^{tx} e^{-\frac{x^2}{2}} dx = C \int_R e^{-\frac{x^2}{2} +tx} dx = e^{\frac{t^2}{2}} C \int_R e^{-\frac{(x-t)^2}{2}}dx

And we have that the moment generating function can be write like this:

\phi(t) = e^{\frac{t^2}{2}

And we can write this as an infinite series like this:

\phi(t)= 1 +(\frac{t^2}{2})+\frac{1}{2} (\frac{t^2}{2})^2 +....+\frac{1}{k!}(\frac{t^2}{2})^k+ ...

And since this series converges absolutely for all the possible values of tX as converges the series e^2, we can use this to write this expression:

E[e^{tX}]= E[1+ tX +\frac{1}{2} (tX)^2 +....+\frac{1}{n!}(tX)^n +....]

E[e^{tX}]= 1+ E[X]t +\frac{1}{2}E[X^2]t^2 +....+\frac{1}{n1}E[X^n] t^n+...

and we can use the property that the convergent power series can be equal only if they are equal term by term and then we have:

\frac{1}{(2k)!} E[X^{2k}] t^{2k}=\frac{1}{k!} (\frac{t^2}{2})^k =\frac{1}{2^k k!} t^{2k}

And then we have this:

E[X^{2k}]=\frac{(2k)!}{2^k k!}, k=0,1,2,...

And then we can find the E[X^2]

E[X^2]= \frac{2!}{2^1 1!}= 1

And we can find the variance like this :

Var(X^2) = E[X^4]-[E(X^2)]^2

And first we find:

E[X^4]= \frac{4!}{2^2 2!}= 3

And then the variance is given by:

Var(X^2)= 3-(1)^2 =2

You might be interested in
A store is having a 20% off sale. If the
Aleks [24]

Answer:

$64.32

Step-by-step explanation:

What you do is:

$53.60×20% → $53.60×0.2

$53.60×0.2=$10.72

$10.72 is the amount of money taken off because of the sale.

Then what you do is:

$53.60+$10.72=_____

53.60

<u>+10.72</u>

64.32

The original price was $64.32.

Hope this helps! Please mark my answer brainliest!

3 0
3 years ago
Read 2 more answers
Shiloh is 7 years older than Courtney. Shiloh's age is 13 years less than two times Courtney's age. The system below models the
Andrews [41]

I'm afraid I don't see the models you have written about although I do know the answer. (C= Courtney's age, S= Shilo's age)

Shilo is 7 years older then Courtney, which equals to C+7=S.

While at the same time Shilo is 13 years younger then twice Courtney's age. This turns into Cx2-13=S.

c+7=cx2-13. This is the equation, now lets work it out.

Add 13 on each side and you get

c+20=2c

Now we subtract c from each side and we get

20=c Therefor we have found out that Courtney is 20 years old.

Shilo being 7 years older then Courtney is 27


8 0
3 years ago
A coral reef grew 3000 millimeters taller. How much did it grow in meters?
ANTONII [103]
3 meters equals 3000 mm
3 0
3 years ago
Can someone help with this question? Thank u! :)
MatroZZZ [7]

Answer:

2.50

Step-by-step explanation:

the pattern shown in the miles driven goes up each time by 5000, stating it's going up by a constant of 5000. the cost goes up each time by 2000.

divide 5000/2000 which would give you 2.5

round it by two decimal places which gives you 2.50

7 0
3 years ago
Need help will mark Brainliest
Kisachek [45]
I believe it is x=55. I could be wrong lol
6 0
3 years ago
Read 2 more answers
Other questions:
  • I will give brainliest to first (2x^2 − 7x^3 + 8x) + (−8x^3 − 3x^2 + 4)
    15·2 answers
  • ?/6 &lt; 1/2 which one is it
    15·2 answers
  • Onnie measures an object's mass in kilograms.
    10·2 answers
  • Hellllllllllllllllp plz...
    6·1 answer
  • [ tan2 θ ] – [( sec θ – 1 )( sec θ + 1 )
    15·1 answer
  • A small class has 9 students, 3 of whom are girls and 6 of whom are boys. The teacher is going to choose two of the students at
    11·1 answer
  • 1Which angle measures less than 90° but greater than 0?
    6·2 answers
  • On the first 7 statistics tests of the semester, Jamal
    7·1 answer
  • What is 9.773 divided by 0.12
    5·2 answers
  • if a rectangle's length is 2 x − 1 and the width is 4 x − 2 , write an expression for the perimeter and an expression for the ar
    13·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!