I know you said "without making any assumptions," but this one is pretty important. Assuming you mean [tex]\alpha,\beta[/tex] are shape/rate parameters (as opposed to shape/scale), the PDF of [tex]X[/tex] is
[tex]f_X(x) = \dfrac{\beta^\alpha}{\Gamma(\alpha)} x^{\alpha - 1} e^{-\beta x} = \dfrac{2^8}{\Gamma(8)} x^7 e^{-2x}[/tex]
if [tex]x>0[/tex], and 0 otherwise.
The MGF of [tex]X[/tex] is given by
[tex]\displaystyle M_X(t) = \Bbb E\left[e^{tX}\right] = \int_{-\infty}^\infty e^{tx} f_X(x) \, dx = \frac{2^8}{\Gamma(8)} \int_0^\infty x^7 e^{(t-2) x} \, dx[/tex]
Note that the integral converges only when [tex]t<2[/tex].
Define
[tex]I_n = \displaystyle \int_0^\infty x^n e^{(t-2)x} \, dx[/tex]
Integrate by parts, with
[tex]u = x^n \implies du = nx^{n-1} \, dx[/tex]
[tex]dv = e^{(t-2)x} \, dx \implies v = \dfrac1{t-2} e^{(t-2)x}[/tex]
so that
[tex]\displaystyle I_n = uv\bigg|_{x=0}^{x\to\infty} - \int_0^\infty v\,du = -\frac n{t-2} \int_0^\infty x^{n-1} e^{(t-2)x} \, dx = -\frac n{t-2} I_{n-1}[/tex]
Note that
[tex]I_0 = \displaystyle \int_0^\infty e^{(t-2)}x \, dx = \frac1{t-2} e^{(t-2)x} \bigg|_{x=0}^{x\to\infty} = -\frac1{t-2}[/tex]
By substitution, we have
[tex]I_n = -\dfrac n{t-2} I_{n-1} = (-1)^2 \dfrac{n(n-1)}{(t-2)^2} I_{n-2} = (-1)^3 \dfrac{n(n-1)(n-2)}{(t-2)^3} I_{n-3}[/tex]
and so on, down to
[tex]I_n = (-1)^n \dfrac{n!}{(t-2)^n} I_0 = (-1)^{n+1} \dfrac{n!}{(t-2)^{n+1}}[/tex]
The integral of interest then evaluates to
[tex]\displaystyle I_7 = \int_0^\infty x^7 e^{(t-2) x} \, dx = (-1)^8 \frac{7!}{(t-2)^8} = \dfrac{\Gamma(8)}{(t-2)^8}[/tex]
so the MGF is
[tex]\displaystyle M_X(t) = \frac{2^8}{\Gamma(8)} I_7 = \dfrac{2^8}{(t-2)^8} = \left(\dfrac2{t-2}\right)^8 = \boxed{\dfrac1{\left(1-\frac t2\right)^8}}[/tex]
The first moment/expectation is given by the first derivative of [tex]M_X(t)[/tex] at [tex]t=0[/tex].
[tex]\Bbb E[X] = M_x'(0) = \dfrac{8\times\frac12}{\left(1-\frac t2\right)^9}\bigg|_{t=0} = \boxed{4}[/tex]
Variance is defined by
[tex]\Bbb V[X] = \Bbb E\left[(X - \Bbb E[X])^2\right] = \Bbb E[X^2] - \Bbb E[X]^2[/tex]
The second moment is given by the second derivative of the MGF at [tex]t=0[/tex].
[tex]\Bbb E[X^2] = M_x''(0) = \dfrac{8\times9\times\frac1{2^2}}{\left(1-\frac t2\right)^{10}} = 18[/tex]
Then the variance is
[tex]\Bbb V[X] = 18 - 4^2 = \boxed{2}[/tex]
Note that the power series expansion of the MGF is rather easy to find. Its Maclaurin series is
[tex]M_X(t) = \displaystyle \sum_{k=0}^\infty \dfrac{M_X^{(k)}(0)}{k!} t^k[/tex]
where [tex]M_X^{(k)}(0)[/tex] is the [tex]k[/tex]-derivative of the MGF evaluated at [tex]t=0[/tex]. This is also the [tex]k[/tex]-th moment of [tex]X[/tex].
Recall that for [tex]|t|<1[/tex],
[tex]\displaystyle \frac1{1-t} = \sum_{k=0}^\infty t^k[/tex]
By differentiating both sides 7 times, we get
[tex]\displaystyle \frac{7!}{(1-t)^8} = \sum_{k=0}^\infty (k+1)(k+2)\cdots(k+7) t^k \implies \displaystyle \frac1{\left(1-\frac t2\right)^8} = \sum_{k=0}^\infty \frac{(k+7)!}{k!\,7!\,2^k} t^k[/tex]
Then the [tex]k[/tex]-th moment of [tex]X[/tex] is
[tex]M_X^{(k)}(0) = \dfrac{(k+7)!}{7!\,2^k}[/tex]
and we obtain the same results as before,
[tex]\Bbb E[X] = \dfrac{(k+7)!}{7!\,2^k}\bigg|_{k=1} = 4[/tex]
[tex]\Bbb E[X^2] = \dfrac{(k+7)!}{7!\,2^k}\bigg|_{k=2} = 18[/tex]
and the same variance follows.
Jyoti is four times older than her daughter after 5 years Jyoti age will be three times of the age of her daughter find their present ages.
Answer:
possible answer: 44- parent/ 14 yrs old daughter.
Sorry if i'm wrong
Step-by-step explanation:
Needs answer please
Answer:
Your answer is.....
2.4 pound in each bag
Mark it as brainlist answer . follow me for more answer.
Step-by-step explanation:
Answer:
Between [tex]2\frac{1}{3} and 2\frac{3}{5}[/tex]
Step-by-step explanation:
12 ÷ 5 = 12/5 = [tex]2\frac{2}{5}[/tex]
find the height of a parallelogram with a base 24 inches and the area of 432 square inches
Answer:
the height of the parallelogram would be 18 inches.
What is the value (3/4)3
Answer:
9/4
Step-by-step explanation:
Answer:
-(11/4)
Step-by-step explanation:
Find the value of 17
33 rounded to the nearest ten-thousandth.
Answer:
0.5152
Step-by-step explanation:
One fifth of the sum of 3 and a number
Answer:
1/5(x+3)
Step-by-step explanation:
That is, assuming the number=x
Write the measure 29° 6'6" as a decimal to the nearest thousandth.
a. 29.1020
C. 29.485
b. 27.5949
d. 27.682°
Please select the best answer from the choices provided
Answer:
A. 29.1020
Step-by-step explanation:
Rewrite it:
29°6'6"
29° + 6(1/60)° + 6(1/3600)°
Simplify:
29° + 0.1° + 0.002°
= 29.102
So, the correct answer is A, 29.1020
HELP
In a fruit basket there are 9 bananas, 4 apples, and 3 plums. Fill in the blanks- The ratio of bananas to apples is ___:___
Answer:
54
Step-by-step explanation:
Judy will use 2 1/2 cups of flour for each cake she bakes. Judy has 10 cups of flour. What is the maximum number of cakes Judy can bake?
Answer:
4 cakes
Step-by-step explanation:
An 8ft length of 4 inch wide crown molding costs $15. How much will it cost to buy 72 ft of crown molding?
Answer:
Answer: $70
EXPLANATION
Step-by-step explanation:
Given, for a 4 inch wide crown molding:
8 ft length costs $14
Cost of 40 ft length = ?
8 ft length costs $14
To find the cost of 1 ft length,
Divide through by 8
8ft/8 length costs $14/8
1ft costs $1.75
Now, to find the cost of 40 ft length
1ft costs $1.75
1 ft x 40 would cost $1.75
40 ft would cost $70
Therefore, it would cost $70 to buy 40 ft of crown molding
In a sample of 310 people, 186 completed only high school, 24 completed only some college, 93 completed a two-year or four-year college, and 7 attended graduate school. What proportion of the sample does not have a two-year or four-year college degree
Answer:
The proportion of the sample that does not have a two-year or four-year college degree is 0.7355.
Step-by-step explanation:
Sample of 310 people.
186(only high school) + 42(only some college) did not have have a two-year or four-year college degree.
So
[tex]p = \frac{186+42}{310} = 0.7355[/tex]
The proportion of the sample that does not have a two-year or four-year college degree is 0.7355.
16 divided by 25 as a decimal
Answer:
The answer of the question is 0.64
Answer:
0.64 is a decimal and 64/100 or 64% is the percentage for 16/25.
Step-by-step explanation:
It will take you to how you do it scan on phone
please help me asap!!!
Answer:
P = 37 units
Step-by-step explanation:
Write the exact value of the side length of a square with an area of
95 square units
The side length of the square is a = 9.75 units
What is the area of a square?A square is a four sided quadrilateral with all of the side having equal lengths. The total internal angle of a square is 360° with each side perpendicular to each other.
A square is a 4 sided polygon where all the sides are of equal length.
The area of the square is the square of any one of its side
Let a be the side of the square
Area of the square = a²
Given data ,
Let the area of the square be represented as A
Now , the value of A is
Area of the square = a²
The exact value of the side length of a square with an area of 95 square units can be found by taking the square root of 95.
√95 ≈ 9.74679434481 (approximate value)
Hence , the exact value of the side length of the square is approximately 9.74679434481 units
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4. A movie-streaming service offers 12 children's movies. This is 15% of the
total number of movies offered. How many movies does the streaming
service offer? *
Answer:
80
Step-by-step explanation:
12 is 15% of 80.
80 movies does the streaming service offer.
What is Algebra?A branch of mathematics known as algebra deals with symbols and the mathematical operations performed on them.
Variables are the name given to these symbols because they lack set values.
In order to determine the values, these symbols are also subjected to various addition, subtraction, multiplication, and division arithmetic operations.
Given:
A movie-streaming service offers 12 children's movies.
This is 15% of the total number of movies offered.
let the total number of movies offered be x.
So, 15% of x= 12
15/100 x = 12
x= 1200/15
x= 80
Thus, there were total 80 movies.
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How are you today?
Take some time to reflect on your day!
Have a great day!
Answer:
I am good!
Step-by-step explanation:
Have a fantastic day!
Answer:
good
Step-by-step explanation:
1
4 adult elephants weigh
6
tons. How much does just one of the elephants weigh?
Answer: 86
Step-by-step explanation:
14 times 6 = 86
Zher and Gabbie are making cookies for the Hospitable Pantry. They only have up to 6 hours to
bake! They can bake 32 chocolate chip cookies in an hour (x), and 20 snickerdoodle cookies in an
hour (y). They need to make at least 150 cookies. Which inequalities represent the situation?
Answer:
32x3=96. 20x3= 60. 60 + 96=156 if you make 3 hours of chocolate chip and 3 hours snickerdoodles you will have 156 cookies
overnight a pipe cracked and started to leak into a basement. the graph shows the depth of the water in relation to time. how many inches does the water rise each hour?
Answer:
if a water pipe bursts over night depends on the pressure of the water but i will say a foot or more an hour
Step-by-step explanation:
sreg
Triangle ABC is reflected across line m to form triangle PRS.
B
R
m
с
Which statement must be true?
O The length of AB = the length of SR
O The length of BC = the length of RP
OMZA = MZS
OmZB = mZR
Answer:
m∠b = m∠r
Step-by-step explanation:
The triangle was reflected across line m.
A reflection is a rigid transformation. A rigid transformation does not change the measurement or size of a shape, so the pre-image and the image are congruent.
The angles on the pre-image corresponds to the angles on the image. This also is for line segments.
This means (for angle measurements):
Angle A is congruent to Angle P
Angle C is congruent to Angle S
Angle B is congruent to Angle R
Option D should be the correct answer.
Find the area of the trapezoid with bases 5cm and 2cm and height 1/5 cm.
(Simplify your answer)
Answer:
1
Step-by-step explanation:
[tex]\frac{(5 + 2) * \frac{1}{5} }{2}[/tex] = [tex]\frac{\frac{7}{5}}{2}[/tex] = [tex]\frac{1.4}{2}[/tex] = 0.7, rounded to the nearest ones is 1.
Answer:
0.7
Step-by-step explanation:
The formula for the area of a trapezoid is (A+B)÷2×H
So our formula is (5+2)÷2×1/5
1.
5+2=7
2.
7÷2=3.5
3.
3.5×1/5=0.7
That's why the answer is 0.7
Diana received an inheritance of $3500 and plans to place all the money in a savings account that pays 3% compounded quarterly. How much money will be in the account in 10 years ?
Answer:
$36,050
Step-by-step explanation:
If the original price Diana puts into the savings account is $3500 and the pays 3% increase, than how much will be in the account in 10 years?
3% can also be turned in 3 divided by 100, I will be using that in my explanation.
3 divided by 100 : 0.03
you then multiply 0.03 by 3500 and you will get 105.
3% of the $3,500 is $105
$3,500 plus $105 is $3,605
You then have to multiply that answer by 10(years)
$3,605 times 10 is $36,050
PLEASE HURRY ASAP:
Which of the following statements are true about the graph of f(x) = 6(x + 1)2 -9?
Check all of the boxes that apply.
The vertex is (1, -9).
The graph opens upward.
The graph is obtained by shifting the graph of f(x) = 6(x + 1)2 up 9 units.
The graph is steeper than the graph of f(x) = x2.
The graph is the same as the graph of f(x) = 6x2 + 12x - 3.
Answer:
loong d is it now we know lololololololol
Step-by-step explanation:
Answer:
2,4,5
Step-by-step explanation:
For what value does (1/9)^a+1=81^a+1 12^2-a?
Answer:
a = -4
Step-by-step explanation:
[tex] (\dfrac{1}{9})^{a + 1} = 81^{a + 1} \cdot 27^{2 - a} [/tex]
[tex] (3^{-2})^{a + 1} = (3^4)^{a + 1} \cdot (3^3)^{2 - a} [/tex]
[tex] 3^{-2a - 2} = 3^{4a + 4} \cdot 3^{6 - 3a} [/tex]
[tex] 3^{-2a - 2} = 3^{4a + 4 + 6 - 3a} [/tex]
[tex] 3^{-2a - 2} = 3^{a + 10} [/tex]
[tex] -2a - 2 = a + 10 [/tex]
[tex] -3a = 12 [/tex]
[tex] a = -4 [/tex]
Which sign makes the statement true? 1
The sign that makes the statement true is < (less than): 1/3 < 2/3.
What is inequality?In mathematics, an inequality is a mathematical expression that shows the relationship between two values or quantities, usually indicating which value is greater than, less than, or equal to the other. Inequalities are represented by the symbols < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to). For example, x < y means x is less than y, and x ≥ y means x is greater than or equal to y. Inequalities can also involve variables, and can be solved to find the range of values that make the inequality true.
Here,
The statement 1/3 < 2/3 means that the value of 1/3 is less than the value of 2/3. In other words, when we compare the two fractions, we find that 2/3 is larger than 1/3.
To understand this better, we can think of a whole object (e.g. a pizza) that we want to divide into three equal parts. If we take only one of those parts, that would be represented by the fraction 1/3. On the other hand, if we take two of those equal parts, that would be represented by the fraction 2/3. Therefore, 2/3 is a larger fraction than 1/3, which is why the sign < (less than) makes the statement true.
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Gwen read the first 15 pages of a book in a bookstore and decided to buy it. Since she bought it, she has been reading 30 pages per day. For which of these functions does y represent the total number of pages Gwen has read if x represents the number of days since she bought the book?
A. x = 15y + 30
b. x = 30y + 15
c. y = 15x +30
d. y = 30x +15
the answer is d. y=30x+15
I need help with this question
Answer:
option B 4√5 is the correct answer of this question .
plz mark my answer as brainlist plzzzz if you find it useful.
Hope this will be helpful to you ☺️.
Answer:
B
Step-by-step explanation:
easy
2x5 is 10, 3x5 is 15
15+10 = 25
25-5 = 20
4x5=20
please solve ive been asking this question so much
Answer: 30
Step-by-step explanation:
✨6 grade math✨ Step by step pls
Answer:
pemdas!!
p=( ), [, ]
e= exponts
m and d= multiplaction and division
a and s = add and subtract
now 3x(3+12/3)-4
= 3x(3+4)/4
= 3x(7)/4
= 3x7/4
= 21-4
= 17
Step-by-step explanation:
Step-by-step explanation:
[tex]3 \times (3 + 12 \div 3) - 4 \\ \\ = 3 \times (3 + 4) - 4 \\ \\ = 3 \times 7 - 4 \\ \\ = 21 - 4 \\ \\ = 17[/tex]
Please hurry and answer
Answer:
B. cot AStep-by-step explanation:
BECAUSE
COS A ÷ SIN A = COT AMARK ME AS BRAINLIST