Answer:
Let x be the amount (in liters) of the 5% acetic acid solution to be added.
Let y be the amount (in liters) of the 10% acetic acid solution to be added.
The system of equations is:
x + y = 2 (the total amount of solution is 2 liters)
0.05x + 0.10y = 0.07(2) (the amount of pure acetic acid in the final solution is 7% of 2 liters)
Simplifying the second equation:
0.05x + 0.10y = 0.14
5x + 10y = 14 (multiplying both sides by 100)
We now have a system of two equations with two variables. We can solve it by substitution or elimination. Here, we'll use substitution:
x + y = 2 => x = 2 - y (subtracting y from both sides)
Substituting x in the second equation:
5(2 - y) + 10y = 14 (replacing x by 2 - y)
10 - 5y + 10y = 14
5y = 4
y = 0.8
So the photographer should add 0.8 liters of the 10% acetic acid solution and 2 - 0.8 = 1.2 liters of the 5% acetic acid solution. We can check that this solution satisfies both equations:
0.05(1.2) + 0.10(0.8) = 0.06 + 0.08 = 0.14
1.2 + 0.8 = 2
Therefore, the answer is:
The photographer should add 0.8 liters of the 10% acetic acid solution and 1.2 liters of the 5% acetic acid solution.
Simplify the following rational expression: \( \frac{500 x^{3}-108}{40 x^{2}+56 x-48} \) Leave the numerator and denominator in your final answer in fac torm.
The numerator is \((25x^{2}+15x+9)\) and the denominator is \(2(x+2)\).
To simplify the given rational expression, we need to factor the numerator and denominator and then cancel out any common factors.
First, let's factor the numerator:
\(500x^{3}-108=4(125x^{3}-27)=4(5x-3)(25x^{2}+15x+9)\)
Next, let's factor the denominator:
\(40x^{2}+56x-48=8(5x^{2}+7x-6)=8(5x-3)(x+2)\)
Now, we can cancel out the common factor of \(4(5x-3)\) from the numerator and denominator:
\(\frac{4(5x-3)(25x^{2}+15x+9)}{8(5x-3)(x+2)}=\frac{(25x^{2}+15x+9)}{2(x+2)}\)
Therefore, the simplified rational expression is \(\frac{(25x^{2}+15x+9)}{2(x+2)}\).
In this simplified form, the numerator is \((25x^{2}+15x+9)\) and the denominator is \(2(x+2)\).
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Which one is it ? please helpppp
The given relation that is a function is the fourth graph which shows a U- shaped curve.
What kind of function is a U- shaped curve ?The graph of a quadratic function is a U-shaped curve called a parabola, which can either face upwards or downwards depending on the value of "a". If "a" is positive, the parabola faces upwards, and if "a" is negative, the parabola faces downwards.
The coefficient "a" determines the width and direction of the parabola, while the constants "b" and "c" determine its position on the coordinate plane. This therefore shows that the fourth graph is a function.
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I am thinking of a number in the hundredths. My number is greater than 0. 8 and less than 0. 9. The greatest digit in the place is in the hundredths place. What number am I thinking of?
The only number that satisfies all the given conditions is 0.88, and this is the number that the person is thinking of.
Based on the given clues, we know that the number is greater than 0.8 and less than 0.9 and that the greatest digit is in the hundredth place. This means that the number must have a digit in the hundredth place that is greater than any digit in the tenth place or the one's place.
To find the number, we can start by considering the largest possible digit in the hundredth place, which is 8. This would give us the number 0.88, which satisfies the condition of being greater than 0.8 and less than 0.9, and also has the greatest digit in the hundredth place.
However, we also need to check if there are any other numbers that satisfy the given conditions. If we consider any digit less than 8 in the hundredth place, we would get a number that is less than 0.88, which is not greater than 0.8. On the other hand, if we consider any digit greater than 8 in the hundredth place, we would get a number that is greater than 0.89, which is not less than 0.9.
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Two trains, Train A and Train B, weigh a total of 437 tons. Train A is heavier than Train B. The difference of their weights is 415 tons. What is the weight of each train?
Step-by-step explanation:
A + B = 437
A - B = 415 (for that sequence it is important to know that A is larger than B).
A = 415 + B
that we use now in the first equation :
415 + B + B = 437
2B = 22
B = 11
A = 415 + B = 415 + 11 = 426 tons
B = 11 tons
Johnny is the Senior Class President and wants to buy Senior Class T-Shirts. The T-
Shirt company is offering the following deal:
-$25 per shirt if you buy 40 shirts or less
- $20 per shirt if you buy more than 40 shirts and up to 100 shirts
-$15 per shirt if you buy more than 100 shirts
Create a Piecewise Function f(x) that you can use to determine the total price of the
shirts. Let x be the number of shirts you order. Use the inequality buttons below to
help you input the Domains for B,D, and F.
A.
B.
C.
D.
E.
F.
A
f(z)= C
if B
if D
E if F
G. What is the cost if you purchased 40 shirts?
Answer:
Step-by-step explanation:
i don’t know
Answer:i really don´t know hopefully you get thw\e answer correct.
Step-by-step explanation:
PLEASE HELP AGAIN.. It takes Adrian 15 minutes to run around the track one time. He runs around the track with Mateo. It takes 30 minutes for both boys to return to the starting point at the same time. If mateo runs faster than Adrian, how long does it take Mateo to run around the track once, assuming that Adrian runs at the same speed as before?
Mateo takes 20 minutes to run around the track once.
What is speed?The rate at which an object goes from one location to another is known as its speed. It is described as the distance that an object covers in a certain amount of time. The distance traveled in one unit of time is typically used to express speed, such as meters per second (m/s), kilometers per hour (km/h), or miles per hour (mph) (mph).
[tex]Speed = \frac{distance}{time }[/tex]
Here we assume that the distance around the track is 'd', Adrian's speed is 'a' (in distance per minute) and Mateo's speed is 'm' (also in distance per minute).
Since Adrian takes 15 minutes to run around the track once, we have:
a = d/15
Let's use the formula: time = distance/speed
to write two equations for the total distance traveled by each boy.
For Adrian: 2d = a × 30
For Mateo: 2d = m × t
where 't' is the time it takes Mateo to run around the track once.
Since we know that Mateo runs faster than Adrian, we have:
m > a
Substituting the expression we found for 'a' into the equation for Adrian, we get:
2d = (d/15) × 30
d = 225
Now, we can solve for Mateo's speed using the equation for the total distance traveled by Mateo:
2d = m × t
2(225) = m × t
m = 450/t
Substituting this expression for 'm' into the inequality m > a, we get:
450/t > d/15
15×450 > d×t
6750 > d×t
Substituting d = 225, we get:
6750 > 225t
t < 30
Since Mateo takes less than 30 minutes to run around the track once, and we know that Adrian takes 15 minutes, the possible value for t is 20 minutes: t = 20.
Therefore, Mateo takes 20 minutes to run around the track once.
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CEREAL A company claims that its boxes of cereal contain 19.1 ounces of
cereal. The actual amount of cereal in the box can be within 0.5 ounce of the
advertised amount. What is the range of possible amounts of cereal c in a box?
A deli owner has room for 65 containers of shredded Parmesan cheese. He has 5-oz and 10-oz containers, and a total of 400 oz of cheese. If 5-oz containers sell for $7 and 10-oz containers sell for $13, how many of each should he sell to maximize his revenue? What is his maximum revenue?
He should sell 50 5-oz containers and 15 10-oz containers to maximize his revenue.
What is cοntainer?A cοntainer is a standard unit οf sοftware that packages up cοde and all its dependencies sο the applicatiοn runs quickly and reliably frοm οne cοmputing envirοnment tο anοther. Cοntainers are a sοlutiοn tο the prοblem οf hοw tο get sοftware tο run reliably when mοved frοm οne cοmputing envirοnment tο anοther. This cοuld be frοm a develοper’s laptοp tο a test envirοnment, frοm a staging envirοnment intο prοductiοn, and perhaps frοm a physical machine in a data center tο a virtual machine in a private οr public clοud.
Corner point (x, y) Value of 8x+14y
0(0,0) 0
A(65,0) 520
B(50, 15) 610
C(0,40) 560
So, the maximum value is 610
Therefore, the maximum point is B(50, 15)
∴ x = 50, y = 15
So, the maximum value is
Therefore, the maximum point is
Hence, he should sell 50 5-oz containers and 15 10-oz containers to maximize his revenue.
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Describe the sequence that you’ll map triangle ABC to A’B’C.
Answer:
First we would shrink by a factor of 3
Then translate 6 units on the x axis
Then reflect across x=6
PLSSSS HELP IF YOU TURLY KNOW THISSS
Answer:
[tex] \sf \: x = 5[/tex]
Step-by-step explanation:
Now we have to,
→ Find the required value of x.
The equation is,
→ 11 + 2x = 3(x + 2)
Then the value of x will be,
→ 11 + 2x = 3(x + 2)
→ 11 + 2x = 3x + 6
→ 2x - 3x = 6 - 11
→ -x = -5
→ [ x = 5 ]
Hence, the value of x is 5.
[tex] \: [/tex]
To find:-[tex] \textrm{the value of x = ?}[/tex][tex] \: [/tex]
Solution:-[tex] \textrm{11 + 2x = 3( x + 2 )}[/tex][tex] \: [/tex]
[tex]\textrm{11 + 2x = 3x + 6}[/tex][tex] \: [/tex]
[tex]\textrm{2x - 3x = 6 - 11}[/tex][tex] \: [/tex]
[tex]\textrm{-1x = -5}[/tex][tex] \: [/tex]
[tex]\rm{x = \cancel\frac{ - 5}{ - 1} }[/tex][tex] \: [/tex]
[tex]\underline { \boxed{\textrm{\purple{x = 5}}}}[/tex][tex] \: [/tex]
The value of x is 5 !
━━━━━━━━━━━━━━━━━━━━━━━
hope it helps! :)
Use the calculator to approximate the trig function of the given angle. Round answer to the fourth decimal place. sin 58°
Based on the provided information The result of rounding to four integer places is sin 58° ≈ 0.8480.
How would you define "rounding"?Making a number easier while maintaining a value that is near to what it was is known as rounding. Although less precise, the outcome is simpler to use. For instance, 73 is 70 when adjusted to the closest ten, since 70 is closer to 73 than 80 is.
Using a calculator, we can approximate sin 58° as follows:
sin 58° ≈ 0.8480
Rounding to four decimal places, we get:
sin 58° ≈ 0.8480
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Judy bought 1 pound 6 ounces of almonds, 2 pounds 3 ounces of walnuts, and 8 ounces of cashews. What was the total weight of the nuts (in pounds and ounces)?
By answering the above question, we may state that This expression's equation evaluation results in: r = 8.5 ft
What is equation?A mathematical equation is a process that links two statements and indicates equality using the equals sign (=). A formal statement that proves the equality of two calculations is known as an equation in algebra. For instance, the equal sign separates the numbers 3x + 5 and 14 in the equation 3x + 5 = 14. A mathematical formula may be used to understand the link between the two phrases that are inscribed on opposite sides of a letter. Frequently, the logo and indeed the particular software are identical. like in 2x - 4 = 2, for example.
The following is the formula for a circle's circumference:
C = 2πr
where is a mathematical constant that is roughly equivalent to 3.14159, C is the circumference, r is the radius, and.
The circle's circumference is stated to be 17 feet. In order to find r, we may put this equal to the circumference formula as follows:
17π = 2πr
When we divide both sides by 2, we obtain:
r = 17π / (2π)
Units cancel out, leaving:
r = 17 / 2
This expression's evaluation results in:
r = 8.5 ft
As a result, the circle's radius is 8.5 feet.
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A sports camp places it's participants on teams. Each team will have 8 members. There are 3208 people registered to participate.
The number of teams that can be formed is 401
Finding the number of teams:
To determine the number of teams that can be formed with the given number of participants, we need to divide the total number of participants by the number of members per team.
Here we have
Number people registered = 3208
Number of members each team will have is 8 members
The number of teams that can be formed is equal to 3208 ÷ 8
Number of teams = 3208/8 = 401
Therefore,
The number of teams that can be formed is 401
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Select all the equations that are equovalent to 52= -4 (2n +1)
The equation that is equivalent to 52 = -4 (2n + 1) is C) -52/4 = 2n + 1.
Starting with the given equation:
52 = -4 (2n + 1)
First, we can simplify the right-hand side of the equation by distributing the -4:
52 = -8n - 4
Then, we can isolate the variable (n) by adding 4 to both sides of the equation:
56 = -8n
Finally, we can solve for n by dividing both sides by -8:
-7 = n
Therefore, we have found that the solution to the equation 52 = -4 (2n + 1) is n = -7.
Now, let's check each of the answer choices to see if they are equivalent to this solution:
A) 13 = -2 (2n + 1)
If we simplify the right-hand side of this equation, we get:
13 = -4n - 2
Then, if we add 2 to both sides and divide by -4, we get:
-3.75 = n
This solution is not equivalent to n = -7, so this equation is not equivalent to the original equation.
B) -7 = -2 (2n + 1)
If we simplify the right-hand side of this equation, we get:
-7 = -4n - 2
Then, if we add 2 to both sides and divide by -4, we get:
1.25 = n
This solution is not equivalent to n = -7, so this equation is not equivalent to the original equation.
C) -52/4 = 2n + 1
If we simplify the left-hand side of this equation, we get:
-13 = 2n + 1
Then, if we subtract 1 from both sides and divide by 2, we get:
-7 = n
This solution is equivalent to n = -7, so this equation is equivalent to the original equation.
D) -13 = -4 (2n + 1)
If we simplify the right-hand side of this equation, we get:
-13 = -8n - 4
Then, if we add 4 to both sides and divide by -8, we get:
1.875 = n
This solution is not equivalent to n = -7, so this equation is not equivalent to the original equation.
Therefore, the equation that is equivalent to 52 = -4 (2n + 1) is C) -52/4 = 2n + 1.
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Find the area of the shaded region under the standard normal curve. If convenient, use technology to find the area. 202 The area of the shaded region is Round to four decimal places as needed)
The area of the shaded region under the standard normal curve is 0.4783.
What is standard normal curve?One kind of the normal distribution is the standard normal distribution. It happens when a typical random variable's mean and standard deviation are both one. In other terms, the term "standard normal distribution" refers to a normal distribution with a mean of 0 and a standard deviation of 1. Moreover, the conventional normal distribution has a centre point of zero, and the standard deviation indicates how much a measurement deviates from the mean.
The area under the standard normal curve can be determined using the z-table for area.
For the given figure the area under the standard normal curve is given as:
P[-2.02<Z<0]
=0.5-0.0217
=0.4783
Hence, the area of the shaded region under the standard normal curve is 0.4783.
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The complete question is:
A dealer sold a car to a man and made a profit of 15percent .The man then sold it to a woman for 120175 naira at a loss of 50percent . how much did the dealer buy the car
The dealer bought the car, which he sold to a man and made a profit of 15 percent at $209,000.
How is the dealer's purchase price determined?To compute the dealer's purchase price, we use backward calculations and percentages.
First, the man's purchase price is determined as $240,350 since he incurred a loss of 50% amounting to $120,175.
By equating the proportional loss to the selling price, we can determine the purchase price paid by the man as $240,350.
Since the dealer generated a profit of 15% on $240,350, we determine his cost as $209,000 based on the profit percentage.
The dealer's profit percentage = 15%
The man's loss percentage = 50%
The selling price by the man to the woman = $120,175
The purchase price by the man = $240,350 ($120,175/50%)
The dealer's purchase price = $209,000 ($240,350/1.15)
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Solvex + 4 - 5 = 6.
A. x = -7 and x
-15
B. x = 7 and x = -7
C. x = -7 and x = 15
D. x = 7 and x = -15
Option:-
[tex] \underline{\sf \color{brown}{ B ) x = 7 nd \: x = -7}}[/tex][tex] \: [/tex]
Given:-
[tex] \sf x + 4 - 5 = 6[/tex][tex] \: [/tex]
Solution:-
[tex] \sf \: x + 4 - 5 = 6[/tex][tex] \: [/tex]
[tex] \sf \: x + 4 = 6 + 5[/tex][tex] \: [/tex]
[tex] \sf \: x + 4 = 11[/tex][tex] \: [/tex]
[tex] \sf \: x = 11 - 4[/tex][tex] \: [/tex]
[tex] \boxed { \sf \blue{x = 7}}[/tex][tex] \: [/tex]
or
[tex] \sf \: x + 4 - 5 = 6[/tex][tex] \: [/tex]
[tex] \sf \: x - 1 = 6[/tex][tex] \: [/tex]
[tex] \sf \: x = 6 + 1[/tex][tex] \: [/tex]
[tex] \boxed{ \sf { \color{skyblue}x = 7}}[/tex][tex] \: [/tex]
━━━━━━━━━━━━━━━━━━━━━━━━━━
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(Giving 120 points and brainly!) Select all the expressions that are equivalent to (3^1)^9
A) 3^-3 . 3 ^-3
B) 3^10
C) 3^7. 3^2
D) 3^-5. 3^-2
Answer:
C) [tex]3^7*3^2[/tex]---------------------------
Simplify each expression including the given and compare.
Given expression:
[tex](3^1)^9=3^{1*9}=3^9[/tex]Options simplify as:
A)
[tex]3^{-3}*3^{-3}=3^{-3-3}=3^{-6}[/tex], this is different from the givenB)
[tex]3^{10}[/tex], this is different from the givenC)
[tex]3^7*3^2=3^{7+2}=3^9[/tex], this is same, so equivalentD)
[tex]3^{-5}*3^{-2}=3^{-5-2}=3^{-7}[/tex], this is different from the givenSet up the trigonometric ratio for right triangles that yoa would ase to fisd
x
. Yeu are aar mied ap lad
x
. 1. a. b. Find
x
to the nearest tenth of a degree. Show your wark. Set up the trigonometric ratio for right triang gkes that you would use to find
x
. You are ase aulend to find
x
, 3. a. b. 4 Approximate
x
to the nearest lenth of a degroe. 5. Consider the following right triangle. Set up the trigonometric ratio for right triangles that you would use to find
x
. Then find
x
.
The key to finding the value of x in a right triangle is to choose the appropriate trigonometric ratio based on the sides given and to use a calculator to find the value of x to the nearest tenth of a degree.
To find the value of x in a right triangle, we can use the trigonometric ratios of sine, cosine, and tangent. The ratio we choose depends on the information given in the question and the sides and angles we are trying to find.
To find x to the nearest tenth of a degree, we can use the following steps:
a. Identify the sides of the triangle that are given and the side we are trying to find.
b. Choose the appropriate trigonometric ratio based on the sides given. For example, if we are given the opposite side and the hypotenuse, we would use the sine ratio.
c. Set up the equation and solve for x. For example, if we are using the sine ratio, the equation would be sin(x) = opposite/hypotenuse.
d. Use a calculator to find the value of x to the nearest tenth of a degree.
To find x using the aulend method, we can use the following steps:
a. Identify the sides of the triangle that are given and the side we are trying to find.
b. Use the Pythagorean theorem, a^2 + b^2 = c^2, to find the missing side.
c. Use the appropriate trigonometric ratio to find the value of x.
To approximate x to the nearest tenth of a degree, we can use a calculator to find the value of x and then round to the nearest tenth.
To find x in the given right triangle, we can use the following steps:
a. Identify the sides of the triangle that are given and the side we are trying to find.
b. Choose the appropriate trigonometric ratio based on the sides given.
c. Set up the equation and solve for x.
d. Use a calculator to find the value of x.
Overall, the key to finding the value of x in a right triangle is to choose the appropriate trigonometric ratio based on the sides given and to use a calculator to find the value of x to the nearest tenth of a degree.
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Tangents. of a circle
The required,
19. AB is tangent to circle C.
20. AB is not tangent.
In a right-angled triangle, its sides, such as the hypotenuse, are perpendicular and the base is Pythagorean triplets.
Here,
Form figure 19.
Applying the Pythagorean theorem,
BS² = AS² + AB²
Substitute the value in the above expression.
12² = 7.2² + [2*4.8]² [radius = diameter /2]
144 = 51.84+92.16
144 = 144
Thus, the required AB is tangent to circle C.
Similarly,
In figure 20 AB is not tangent because,
BC² ≠ AB² + AC²
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Area of rectangles. Please help!
Answer:
11.35
or 11 7/20
Step-by-step explanation:
IfA=[aij]us a2×3matrix, such thataij=5(−i+2j)2. Thena12is: Select one:535953656
The value of a12 is 45.
The given matrix A = [aij] is a 2x3 matrix, which means it has 2 rows and 3 columns. The given condition is aij = 5(-i + 2j)^2. We need to find the value of a12, which means the element in the first row and second column.
To find the value of a12, we need to substitute i = 1 and j = 2 in the given condition.
a12 = 5(-1 + 2*2)^2
= 5(3)^2
= 5*9
= 45
Therefore, the value of a12 is 45.
So, the correct option is:
Select one:
a. 53
b. 59
c. 45
d. 56
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a printer makes more than three copies of labels every hour graph the number of books made in 2 hours.
Use substitution to solve
The solution for this system of equations is any ordered pair of the form (x, 3x-3), where x is any real number.
What is Equation?Two or more expressions with an Equal sign is called as Equation.
We can solve this system of equations using substitution. Solving the second equation for y, we get:
y = 3x - 3
Substituting this expression for y into the first equation, we get:
9x - 3(3x - 3) = 9
9x - 9x + 9 = 9
Therefore, the equation is true for any value of x.
Hence, the solution for this system of equations is any ordered pair of the form (x, 3x-3), where x is any real number.
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Hence determine the volume of the Nestle Cremora Carton container in m³
The volume of the Nestle Cremora Carton container is 400 m³
What is an equation?An equation is an expression that shows how two or more numbers and variables are related using mathematical operations of addition, subtraction, multiplication, division, exponents and so on.
The volume of a cuboid is given as:
Volume = Length * Width * Height
The question is incomplete. Let us assume that the container dimensions are as follows:
Height: 10 mLength: 8 mWidth: 5 mThe Nestle Cremora Carton container is in the shape of a cuboid. Hence:
Volume of container = length * width * height = 10 m * 8 m * 5 m = 400 m³
The volume is 400 m³
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Write a second equation whose graph goes thru(0,2) so the system has one solution at (4,1)
Answer:
y= -1/4x+b
Step-by-step explanation:
see image
In a basket 5/6 of the fruits are apples. 4/5 of the apples are red apples. What fraction of the fruits in the basket red apples? Give the answer in simplest form
Answer:
2/3 of the fruits in the basket are red apples.
Step-by-step explanation:
5 apples of 6 total fruit are in a basket.
4 apples are red.
Take 4 and divide it by 6, you will get 0.67 or 2/3.
2/3 cannot be simplified.
ctIONS through a given point with a given slope ope -(2)/(3) passing through the point (3,4)
The equation of the line with slope -(2)/(3) passing through the point (3, 4) is y = -(2)/(3)x + 6. To find the equation of a line that passes through a given point with a given slope, we can use the point-slope formula. The point-slope formula is: y - y1 = m(x - x1), where m is the slope and (x1, y1) is the point that the line passes through. In this case, the slope is -(2)/(3) and the point is (3, 4).
Plugging these values into the point-slope formula, we get:
y - 4 = -(2)/(3)(x - 3)
We can then rearrange the equation to get it into slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept:
y - 4 = -(2)/(3)x + 2
y = -(2)/(3)x + 6
So, the equation of the line with slope -(2)/(3) passing through the point (3, 4) is y = -(2)/(3)x + 6.
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Find the size of angle x. 65° X 35°
The value of 'x' using the Triangle sum theorem is found as: x = 80°.
Explain about the Triangle sum theorem?According to the triangle sum theorem, all of a triangle's inner angles add up to 180 degrees. It is also known as the triangle's angle sum property.The triangle sum principle states that the sum of another three angles of any triangle is 180 degrees. According to their sides and angles, triangles can be classified into various mathematical kinds. These triangles are all three-sided and adhere to the triangle sum concept.Given that the triangle in the image is a triangle and that two of its angles is 65° and 35°.180° = 65°+35°+x
180° = 100°+x
180° - 100° = x
x=80° (value of x)
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Let Py be a discrete distribution on {0,1,2,...} and given Y = y, the conditional distribution of X be the binomial distri- bution with size y and probability p. Show that (i) if y has the Poisson distribution with mean θ, then the marginal distri- bution of X is the Poisson distribution with mean pθ; (ii) if Y + r has the negative binomial distribution with size r and proba- bility π, then the marginal distribution of X +r is the negative binomial distribution with size r and probability π/(1 -(1-P)(1 - 7)).
probability π/(1-(1-p)(1-π)).
It is given that:Let Py be a discrete distribution on {0,1,2,...} and given Y = y, the conditional distribution of X be the binomial distribution with size y and probability p. We need to show that:(i) if y has the Poisson distribution with mean θ, then the marginal distribution of X is the Poisson distribution with mean pθ;(ii) if Y + r has the negative binomial distribution with size r and probability π, then the marginal distribution of X +r is the negative binomial distribution with size r and probability π/(1 -(1-P)(1 - 7)).Solution:Let us consider each part one by one.(i) If y has the Poisson distribution with mean θ, then the marginal distribution of X is the Poisson distribution with mean pθ.We are given that Y = y, the conditional distribution of X be the binomial distribution with size y and probability p.So, P(X = x | Y = y) = yCxpy(1−p)y−x , x = 0,1,2,...,y.Now, we need to find the marginal distribution of X. We have:P(X = x) = ∑y=P(Y=y)P(X=x|Y=y) = ∑y=P(Y=y)yCxpy(1−p)y−xLet us calculate the above sum using the Poisson distribution of y. For this, we have to calculate the probability P(Y=y).We are given that y has the Poisson distribution with mean θ.So, P(Y=y) = e−θθy/y!∑y=0∞P(Y=y)yCxpy(1−p)y−x=∑y=0∞e−θθy/y!yCxpy(1−p)y−x=px∑y=0∞e−θθy−x/y!(y−x)!p(y−x)(1−p)y−x=px∑k=0∞e−θθk/k!(x+k)!(1−p)k=px∑k=0∞(θ(1−p)x(1−p)k/k!(x+k)!)e−θ(1−p)(1−p)kThe above sum is the sum of terms of the form akk! where ak = (θ(1−p)x(1−p)k/k!(x+k)!). Such a sum can be expressed in the form of the Poisson distribution. Thus we get:P(X = x) = ∑y=P(Y=y)P(X=x|Y=y) = px∑k=0∞(θ(1−p)x(1−p)k/k!(x+k)!)e−θ(1−p)(1−p)k= e−pθ∑k=0∞(pθ(1−p)x(1−p)k/k!(x+k)!)e−pθ(1−p)(1−p)k= e−pθ∑k=0∞Pois(pθ)(x+k)k! (1−p)kWe recognize the above sum as the Poisson distribution with mean pθ. Thus we get:P(X = x) = e−pθ(pθ)x/x!The marginal distribution of X is the Poisson distribution with mean pθ.(ii) If Y + r has the negative binomial distribution with size r and probability π, then the marginal distribution of X +r is the negative binomial distribution with size r and probability π/(1 -(1-P)(1 - 7)).Let us first consider the conditional distribution of X given Y = y. We are given that Y + r has the negative binomial distribution with size r and probability π. This means that the sum of y + r independent and identically distributed Bernoulli random variables each with probability p has the negative binomial distribution with size r and probability π.So, P(X = x | Y = y) = (y+r)xpx(1−p)y+r−x, x = 0,1,2,...,y+r.Now, we need to find the marginal distribution of X + r. We have:P(X + r = k) = ∑y=P(Y=y)P(X+r=k|Y=y) = ∑y=P(Y=y)(y+r)kpr(1−p)y+r−kLet us simplify the above sum. For this, we have to calculate the probability P(Y = y).We are given that Y + r has the negative binomial distribution with size r and probability π.So, P(Y = y) = (y + r - 1)Cyπr(1-π)y, y = 0, 1, 2, …Now, we can express the above sum in the form of the negative binomial distribution. Thus we get:P(X + r = k) = ∑y=P(Y=y)(y+r)kpr(1−p)y+r−k= ∑y=P(Y=y)(y+r-1)Cyπr(1-π)ykpπr(1-π)k-y-r+1= NegBin(k-r+r, π/(1-(1-p)(1-π)))The marginal distribution of X + r is the negative binomial distribution with size r and probability π/(1-(1-p)(1-π)).
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