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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

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Find the area if the composite shape
soldi70 [24.7K]

Answer:

111 m²

Step-by-step explanation:

A rectangle is a quadrilateral (has four sides and four angle) with two pairs of parallel sides. Opposite sides of a rectangle are equal to each other. Also all the angles of a rectangle are 90° each.

The area of a rectangle = length * width

For rectangle 1, length = 12 m, width = 3 m

Therefore area of rectangle 1 = length * width = 12 m * 3 m = 36 m²

For rectangle 2, length =(12 m - 3 m - 3 m) = 6 m, width =(15 m - 10 m) =5 m

Therefore area of rectangle 2 = length * width = 6 m * 5 m = 30 m²

For rectangle 3, length = 15 m, width = 3 m

Therefore area of rectangle 3 = length * width = 15 m * 3 m = 45 m²

Area of composite shape = Area of rectangle 1 + Area of rectangle 2 + Area of rectangle 3

Area of composite shape = 36 m² + 30 m² + 45 m² = 111 m²

3 0
3 years ago
Given f of x is equal to 1 over the quantity x minus 3 end quantity and g of x is equal to the square root of the quantity x plu
Lostsunrise [7]

The domain of the composite function is given as follows:

[–3, 6) ∪ (6, ∞)

<h3>What is the composite function of f(x) and g(x)?</h3>

The composite function of f(x) and g(x) is given as follows:

(f \circ g)(x) = f(g(x))

In this problem, the functions are:

  • f(x) = \frac{1}{x - 3}.
  • g(x) = \sqrt{x + 3}

The composite function is of the given functions f(x) and g(x) is:

f(g(x)) = f(\sqrt{x + 3}) = \frac{1}{\sqrt{x + 3} - 3}

The square root has to be non-negative, hence the restriction relative to the square root is found as follows:

x + 3 \geq 0

x \geq -3

The denominator cannot be zero, hence the restriction relative to the denominator is found as follows:

\sqrt{x + 3} - 3 \neq 0

\sqrt{x + 3} \neq 3

(\sqrt{x + 3})^2 \neq 3^2

x + 3 \neq 9

x \neq 6

Hence, from the restrictions above, of functions f(x), g(x) and the composite function, the domain is:

[–3, 6) ∪ (6, ∞)

More can be learned about composite functions at brainly.com/question/13502804

#SPJ1

7 0
1 year ago
Will give out brainliest for the CORRECT answer!
ch4aika [34]

Answer:

162-x=180

x=342

Step-by-step explanation:

4 0
2 years ago
Read 2 more answers
A hollow metallic cylinder is made from a metallic sheet of 2.4 cm thickness. The inner
butalik [34]

Answer:

The volume of the metal used is 5277.9 cm³.

Step-by-step explanation:

The volume of a cylinder is given by:

V = \pi r^{2} h

Where:

r: is the radius

h: is the height = 50 cm

The volume of metal (V_{m}) can be found by subtracting the internal volume (V_{i}) from the external volume (V_{e}):

V_{m} = V_{e} - V_{i}

V_{m} = \pi r_{e}^{2} h - \pi r_{i}^{2} h    

The external radius is given by the sum of the internal radius with the tickness:

r_{e} = t + r_{i}

Hence, the volume is:

V_{m} = \pi*50 cm((2.4 cm + 5.8 cm)^{2} - (5.8 cm)^{2}) = 5277.9 cm^{3}

           

Therefore, the volume of the metal used is 5277.9 cm³.

I hope it helps you!                                                      

4 0
3 years ago
Vonda works between 30 and 32 hours per week at a hair salon. She pays a one time $250 chair rental fee, and earns $40 per hour
Vlad1618 [11]

Answer:

The practical domain of the function is [30,32].

Step-by-step explanation:

The given function is

p(h)=40h-250

where, p(h) represents the Vonda's weekly pay as a function of hours worked.

She pays a one time $250 chair rental fee, and earns $40 per hour that she works.

Domain is the set of all possible inputs.

The possible domain of the given function is all real numbers but the number of hours can not be negative, therefore  h>0.

It is given that Vonda works between 30 and 32 hours per week at a hair salon. So,

30

Therefore practical domain of the function is [30,32].

3 0
3 years ago
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