please help me with math i’ll give you brainlist

The total number of cans in the display is 611.

Let's denote the number of cans in the first row as x.

According to the problem statement, each subsequent row has 5 fewer cans than the row beneath it.

Therefore, the number of cans in the second row will be x - 5, the number of cans in the third row will be x - 10, and so on.

We are given that there are 13 rows in total, and the top row has 1 can. Therefore, we can write the following equation:

x + (x - 5) + (x - 10) + ... + (x - 60) + (x - 65) = 1 + 2 + 3 + ... + 12 + 13

Simplifying the left-hand side, we can combine like terms:

13x - (5 + 10 + 15 + ... + 60 + 65) = 91

Using the formula for the sum of an arithmetic series, we can evaluate the sum of the numbers in parentheses:

5 + 10 + 15 + ... + 60 + 65 = (13/2)(5 + 65) = 455

Substituting this value into the equation, we get:

13x - 455 = 91

Solving for x, we find that:

x = 46

Therefore, the number of cans in each row is:

46, 41, 36, ..., 1

To find the total number of cans, we can use the formula for the sum of an arithmetic series:

n/2(a + l)

where;

In this case, n = 13, a = 46, and l = 1. Plugging in these values, we get:

13/2 x (46 + 1) = 13/2 x 47 = 611.5

Since we can't have a fraction of a can, the total number of cans in the display is 611.

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The coordinates of point X are (7/3, 2√(2)) in Quadrant I.

In coordinate geometry, a plane is divided into four regions called quadrants. These quadrants are numbered counterclockwise starting from the upper right quadrant, which is known as the first quadrant.The four quadrants are defined by the x-axis and y-axis. The x-axis is a horizontal line that runs left and right through the origin, while the y-axis is a vertical line that runs up and down through the origin.

An isosceles triangle is a type of triangle in which two sides are of equal length. In an isosceles triangle, the third side is called the base, and the two equal sides are called legs. The two angles formed by the legs and the base are also equal to each other. The angle opposite the base is called the vertex angle.

In the given question,

Since Triangle XYZ is a 45°- 45°- 90° triangle, the two legs of the triangle are congruent.

Let's call the length of each leg "x".We know that point Z is the right-angle vertex of the triangle and has coordinates (6, 2).

Since the triangle is isosceles, we can find the length of the other leg using the distance formula:x² + x² = (distance between Y and Z)²²x² = (6 - (-1))²²x² = 49x² = 24.5.

Now that we know the length of each leg, we can find the coordinates of point X. Since Triangle XYZ is a 45°- 45°- 90° triangle, we know that the hypotenuse is the square root of 2 times the length of a leg.

Let's call the coordinates of point X (x, y).x² + y² = (distance between X and Y)²x² + y² = x² + (y - 2)²y = 2√(2)Now we can find the x-coordinate of point X:x^2 + (2√(2))^2 = (distance between X and Z)^2x^2 + 8 = (6 - x)^2x^2 + 8 = 36 - 12x + x^212x = 28x = 7/3.

Therefore, the coordinates of point X are (7/3, 2√(2)) in Quadrant I.

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Answer:

Let the four digits be represented by abcd, where a, b, c, and d represent the digits in the thousands, hundreds, tens, and ones places, respectively.

From the first condition, we have:

a + b = c + d

From the second condition, we have:

a + d = 2(c + b)

Simplifying the second condition, we get:

a - 2b + c - d = 0

From the third condition, we have:

10a + b = 2(10c + d)

Simplifying the third condition, we get:

5a - 2b - 5c + 2d = 0

Now we have four equations with four variables. We can use substitution and elimination to solve for the variables.

From the first equation, we have:

a = c + d - b

Substituting into the second equation, we get:

c + d - b + d = 2(c + b)

Simplifying, we get:

2d - 3b + c = 0

From the third equation, we have:

10c + d = 5a

Substituting a with c + d - b, we get:

10c + d = 5(c + d - b)

Simplifying, we get:

5c - 4d + 5b = 0

Now we have two equations with two variables (2d - 3b + c = 0 and 5c - 4d + 5b = 0). Solving for b in terms of c, we get:

b = (5d - c)/3

Substituting into the first equation, we get:

2d - (5d - c)/3 + c = 0

Simplifying, we get:

7c - 13d = 0

Thus, c = 13/7d. Since c is a digit, d must be a multiple of 7. The only possible values for d are 1, 7, and 9.

If d = 1, then c = 13/7, which is not a digit.

If d = 7, then c = 13, b = 2, and a = 18. This satisfies all the conditions, and the four digits in order are 1872.

If d = 9, then c = 18, which is not a digit.

Therefore, the final four digits are 1872.

(please mark my answer as brainliest)

The ROC of a given signal's Z-transform can be determined without actually computing the Z-transform by identifying the maximum and minimum magnitude of the signal and checking for any poles of the Z-transform within the resulting annular region.

Let's take a signal as an example, suppose x[n] = {1, -2, 3, -4, 5}. In order to determine the ROC of its Z-transform, we are firstly required to first look for any regions in the complex plane where the sum of the absolute values of the Z-transform is found finite. It can be done by looking for the maximum and minimum magnitude of x[n] and denote them as R1 and R2 respectively. Then, the ROC of the Z-transform will be the annular region between R1 and R2, excluding any poles of the Z-transform that lie within this annular region.

In this case, the maximum absolute value of x[n] is 5 and the minimum is found being 1. So, the ROC of the Z-transform will be the annular region between |z| = 1 and |z| = 5. We can denote this as 1 < |z| < 5. We also need to check if there are any poles of the Z-transform within this annular region. Since we haven't actually computed the Z-transform, we cannot determine the exact location of any poles.

However, we can check for any values of z that would make the Z-transform infinite. For example, if x[n] is a causal signal (i.e., x[n] = 0 for n < 0), then the ROC cannot include any values of z for which |z| < 1, since this would make the Z-transform infinite.

So, the ROC of the Z-transform for the given signal x[n] can be written as 1 < |z| < 5, assuming that x[n] is a causal signal.

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The complete question is :

Can you explain how to determine the ROCs (regions of convergence) for the Z-transform of a given signal without actually computing the Z-transform? Please provide an example signal with random data and demonstrate how to find its ROCs using this method.

The Nutty Prοfessοr shοuld use apprοximately 14.94 pοunds οf cashews and 35 - 14.94 = 20.06 pοunds οf Brazil nuts tο make a 35 pοund mixture that sells fοr $5.31 per pοund.

Assume the Nutty Prοfessοr makes a 35-pοund mixture with x pοunds οf cashews and (35 - x) pοunds οf Brazil nuts.

The cashews cοst $6.80 per pοund, sο the tοtal cοst οf x pοunds οf cashews is $6.8x dοllars.

Similarly, Brazil nuts cοst $4.20 per pοund, sο (35 - x) pοunds οf Brazil nuts cοst 4.2(35 - x) dοllars.

The tοtal cοst οf the mixture equals the sum οf the cashew and Brazil nut cοsts, which is:

6.8x + 4.2(35 - x) (35 - x)

When we simplify, we get:

6.8x + 147 - 4.2x

2.6x + 147

The mixture sells fοr $5.31 per pοund, sο the tοtal revenue frοm selling 35 pοunds οf the mixture is:

35(5.31) = 185.85

When we divide the tοtal cοst οf the mixture by the tοtal revenue, we get:

2.6x + 147 = 185.85

Subtractiοn οf 147 frοm bοth sides yields:

2.6x = 38.85

When we divide by 2.6, we get:

x ≈ 14.94

Tο make a 35-pοund mixture that sells fοr $5.31 per pοund, the Nutty Prοfessοr shοuld use apprοximately 14.94 pοunds οf cashews and 35 - 14.94 = 20.06 pοunds οf Brazil nuts.

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Answer:

470.4 mm

Step-by-step explanation:

Given: the annual rainfall in 2017 in opuwo was 420 mm, the annual rainfall in 2018 was 12% more than in 2017.

First, find 12% of 420 mm:

12% of 420 mm

[tex]\frac{12}{100}[/tex] x 420 mm

1.2 x 420 mm

= 50.4 mm

Then add 50.4 mm to the previous annual rainfall of 420 mm:

50.4 mm + 420 mm

= 470.4 mm

Therefore, the annual rainfall in opuwo in 2018 is 470.4 mm.

Answer: (f⋅g)(1) = 10.

Step-by-step explanation:

To find (f⋅g)(x), we need to multiply the two functions f(x) and g(x) together. This can be done by multiplying each term of f(x) by each term of g(x), and then combining like terms. We get:

(f⋅g)(x) = f(x) * g(x)

= (-7x+3) * (3x^2 - 4x - 1)

= -21x^3 + 28x^2 + x - 3x^2 + 4x + 1

= -21x^3 + 25x^2 + 5x + 1

To find (f⋅g)(1), we can substitute x=1 into the expression for (f⋅g)(x):

(f⋅g)(1) = -21(1)^3 + 25(1)^2 + 5(1) + 1

= -21 + 25 + 5 + 1

= 10

Therefore, (f⋅g)(1) = 10.

The f_max is the maximum value of the torsional force that can be applied to a shaft without causing it to fail.

To draw torsional moment and displacement diagrams, you can use a process called “torque diagram” which involves solving for all external moments acting on the shaft and drawing out a free body diagram of the shaft horizontally, rotating the shaft if necessary, so that all torques act around the horizontal axis. (Refer the image)

Lined up below the free body diagram, draw a set of axes. The x-axis will represent the location (lined up with the free body diagram above), and the y-axis will represent the internal torsional moment, with positive numbers indicating an internal torsional moment vector to the right and negative numbers indicating an internal torsional moment to the left. The maximum value of torsional moment is denoted by T_max or F_max.

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The segments which is diameter are NG and XR. So correct option is A and C.

In geometry, a segment is a part of a line that is bounded by two distinct endpoints, and it contains every point on the line that is between those endpoints. A segment can be thought of as a straight line that has been cut into two parts, with a specific length. The two endpoints of a segment are typically named using capital letters, such as A and B, and the segment itself is denoted using a line over the two letters, such as AB.

Difference

In geometry, a segment and a diameter are both related to circles, but they are different concepts.

A segment is a part of a circle that is bounded by two points on the circle and a chord connecting those two points. Essentially, a segment is a portion of the circle that is cut off by a line. A circle can have many segments, depending on how the line intersects the circle.

On the other hand, a diameter is a line segment that passes through the center of the circle and has its endpoints on the circle. A diameter is the longest chord of a circle and divides the circle into two equal halves. In other words, the diameter is the distance across the circle, passing through the center.

In summary, a segment is a portion of a circle that is cut off by a line, while a diameter is a line segment that passes through the center of the circle and has its endpoints on the circle.

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The open circle at 3 indicates that 3 is not included in the solution set. This inequality can be read as "X is less than 5" or "X is strictly less than 5."

In mathematics, an expression is a combination of numbers, symbols, and operators that represents a value. Expressions can be as simple as a single number or variable, or they can be complex combinations of mathematical operations. For example, 2 + 3 is a simple expression that represents the value 5, while (2 + 3) x 4 - 1 is a more complex expression that represents the value 19. Expressions can be evaluated or simplified using the rules of arithmetic and algebra.

Here,

1. Simplify:

3(4x-2)+ 7X (2-1) + 4 (6+4)+(-8)

Multiplying inside the parentheses first:

12x - 6 + 7x + 4 + 40 - 8

Combining like terms:

19x + 30

Final answer: 19x + 30

2. Graph:

3 > X

This is a simple inequality in one variable (X). To graph it on a number line, we first draw a dot at 3 (since the inequality is strict), and then shade all values less than 3:

<=========o---

The open circle at 3 indicates that 3 is not included in the solution set.

3. Write the inequality:

X < 5

This is a simple inequality in one variable (X). The inequality sign is "less than," and the number on the right-hand side is 5. This inequality can be read as "X is less than 5" or "X is strictly less than 5."

4. Solve for x:

3x - 7 = 42

Adding 7 to both sides to isolate the variable:

3x = 49

Dividing both sides by 3 to solve for x:

x = 16.33 (rounded to two decimal places)

Final answer: x = 16.3

5. Find 32% of $542.50:

To find 32% of $542.50, we can use the formula:

percent * amount = part

where "percent" is the percentage expressed as a decimal, "amount" is the whole amount, and "part" is the result we're looking for.In this case, we have:

0.32 * $542.50 = part

Multiplying:

$173.60 = part

Final answer: $173.60

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The weight that corresponds to this event are approximately 344.03 grams and 375.97 grams.

To find the weight that corresponds to each event, we need to use the standard normal distribution, which has a mean of 0 and a standard deviation of 1. We can convert the given mean and standard deviation to z-scores using the formula:

z = (x - μ) / σ

where x is the weight we want to find, μ is the mean (360 grams), and σ is the standard deviation (9 grams).

Then, we can use a standard normal distribution table or calculator to find the probability of each event, and convert it back to a weight using the inverse of the z-score formula:

x = μ + z * σ

where z is the z-score that corresponds to the desired probability.

Event 1: The weight is less than 345 grams.

z = (345 - 360) / 9 = -1.67

Using a standard normal distribution table or calculator, we find that the probability of a z-score less than -1.67 is approximately 0.0475.

x = 360 + (-1.67) * 9 = 344.03 grams

Therefore, the weight that corresponds to this event is approximately 344.03 grams.

Event 2: The weight is between 355 and 365 grams.

First, we need to find the z-scores that correspond to the two boundaries:

z1 = (355 - 360) / 9 = -0.56

z2 = (365 - 360) / 9 = 0.56

Using a standard normal distribution table or calculator, we find that the probability of a z-score less than -0.56 is approximately 0.2123, and the probability of a z-score less than 0.56 is approximately 0.7123. Therefore, the probability of a z-score between -0.56 and 0.56 is:

0.7123 - 0.2123 = 0.5

x1 = 360 + (-0.56) * 9 = 355.16 grams

x2 = 360 + (0.56) * 9 = 364.84 grams

Therefore, the weight that corresponds to this event is any weight between 355.16 and 364.84 grams.

Event 3: The weight is greater than 375 grams.

z = (375 - 360) / 9 = 1.67

Using a standard normal distribution table or calculator, we find that the probability of a z-score greater than 1.67 is approximately 0.0475.

x = 360 + (1.67) * 9 = 375.97 grams

Therefore, the weight that corresponds to this event is approximately 375.97 grams.

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a) A function showing the value of the account after t years, where the annual growth rate can be found from a constant, is f(x) = 690 (1+0.0055)^4t.

b) The percentage of growth per year (APY) is 2.2%.

A function is a mathematical expression that shows the relationship between variables.

An example of a mathematical function is an equation that shows the relationship between y and x variables.

Principal = $690

APR = 2.2%

APR per quarter = 0.0055 (2.2%/4)

Compounding = Quarterly

Investment period = t years

Let f(x) = the value of the account after t years.

Future value function, (FV) = PV × (1 + r) ^ n

Where PV = present value or investment

r = compounding rate per period

n = the investment period

Therefore, f(x)  or FV = 690 (1+0.0055)^4t.

APY = 100 [(1 + Interest/Principal)(365/Days in term) - 1]

2.2% = 100 [(1 + $15.18/$690)(365/365) - 1]

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The commission earned 4 percentage on the salesperson on the sale of furnaces is $1320.

In mathematics, a percentage is a number or ratio that can be expressed as a fraction of 100. If we need to calculate a percentage of a number, we should divide it by its entirety and then multiply it by 100. The percentage therefore refers to a part per hundred. The word per cent means per 100. The letter "%" stands for it. The term "percentage" was adapted from the Latin word "per centum", which means "by the hundred". Percentages are fractions with 100 as the denominator.

by the question.

the commission that the salesperson earns on the sale of $33,000 worth of furnaces= 4% of 33,000 = 4× 330 = $1320

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Answer:

2

Step-by-step explanation:

2x-y = -3

differentiate both sides with respect to x

(2x-y)' = (-3)'

2 - dy/dx = 0

dy/dx = 2

If we solve this inequality for x, we get x 20, which indicates the school equation bake sale must sell at least 20 cakes in order to meet their $200 objective.

An equation in mathematics is a statement that states the equality of two expressions. An equation is made up of two sides that are separated by an algebraic equation (=). For example, the argument "2x + 3 = 9" asserts that the phrase "2x + 3" equals the number "9". The purpose of equation solving is to determine the value or values of the variable(s) that will allow the equation to be true. Equations can be simple or complicated, regular or nonlinear, and include one or more elements. In the equation "x2 + 2x - 3 = 0," for example, the variable x is raised to the second power. Lines are utilised in many different areas of mathematics, such as algebra, calculus, and geometry.

10x ≥ 200 accurately portrays the scenario.

To explain why, consider the following:

Because each cake costs $10, the total amount earned from selling x cakes is 10x.

The aim is to raise at least $200, thus the total amount must be larger than or equal to $200.

As a result, we may describe the circumstance by writing the inequality 10x 200.

If we solve this inequality for x, we get x 20, which indicates the school bake sale must sell at least 20 cakes in order to meet their $200 objective.

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Answer:

Example of a function that satisfies the following conditions:

a. Domain and range are both all real numbers except 5:

f(x) =

{

x if x ≠ 5

0 if x = 5

}

b. Domain is all positive numbers greater than 1, including 1:

f(x) =

{

(x-1)^2 / (x-1) if x > 1

0 if x = 1 or x = 5

}

c. Domain is all positive numbers greater than 1, but not including 1:

f(x) =

{

(x-1)^2 / (x-1) if x > 1 and x ≠ 1

0 if x = 1 or x = 5

}

The correct option is C. P, J : 9 units from this origin, points are on the axes.

A pair of numbers called coordinates are used to locate a point or a form in a two-dimensional plane.

Other coordinate types include:

Thus, Points P, J are 9 units from this origin which lies on the negative x axis and negative y axis respectively.

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Complete question is-

Which points are on the axes 9 units from the origin?

A.

D, Z, L

B.

L, P

C.

P, J

D.

J, E

The graph the question is attached.

Answer: .70

Step-by-step explanation:

10^-4 = .0001

.0001 x 9=  .0009

.00063/ .0009= .70

The length of RS is 13 units and the length of the hypotenuse XY is approximately 16.4 units.

15) In ΔTSR

TR² = TS² + RS²

Substituting the given values, we get:

(5√10)² = 9² + RS²

250 = 81 + RS²

RS² = 169

Taking the square root of both sides, we get:

RS = 13 units

16)In ΔYZX

XY² = YZ² + XZ²

Substituting the given values, we get:

XY² = 10² + 13²

XY² = 169 + 100

XY² = 269

Taking the square root of both sides, we get:

XY = √269

XY=16.4 units

What is Pythagorean theorem?

In a right triangle, the Pythagorean theorem states that the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

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The magnitude of the substitution effect depends on the slope of the indifference curve and the specific prices and quantities of the goods U4 has the largest substitution effect. so the correct answer is D).

The substitution effect measures the change in the quantity demanded of a good in response to a change in its relative price, while keeping the consumer's utility constant. It depends on the slope of the indifference curve at the initial consumption bundle.

If the indifference curve is relatively steep like curve U4, then the substitution effect will be larger because a small change in the relative price of the goods will cause the consumer to switch to a consumption bundle that is further away from the original bundle.

On the other hand, if the indifference curve is relatively flat like U1, the substitution effect will be smaller because the consumer can switch between the goods while keeping the utility constant with a relatively small change in the relative prices.

So, the substitution effect will be largest for indifference curve is U4. So, option D) is correct.

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_____The given question is incomplete, the complete question is given below:

The substitution effect will be largest for indifference curve Quantity of good Y 10- 9- 5- 0123 456789 10 Quantity of good X

a) U1

b) U2.

c) U3.

d) U4.

a) 14 to 15, 5-digit hexadecimal are there in which the first digit is E or F

b) 1205 (in decimal) 6-digit hexadecimal start with a letter (A-F) and end with a numeral (0-9)

c) there is a total of 2.1875, 3-digit hexadecimal start with a letter (A-F) or end with a numeral (0-9) (or both)

The hexadecimal number system( hex) functions nearly identically to the decimal and double systems. rather of using a base of 10 or 2 independently, it uses a base of 16. Hex uses 16 integers including 0- 9, just as the decimal system does, but also uses the letters A, B, C, D, E, and F( original to a, b, c, d, e, f) to represent the figures 10- 15. Every hex number represents 4 double integers, called bites, which makes representing large double figures simpler. For illustration, the double value of 1010101010 can be represented as 2AA in hex. This helps computers to compress large double values in a manner that can be fluently converted between the two systems.

a) 14 to 15, 5-digit hexadecimal are there in which the first digit is E or F

b) 4 is in the 16 x 16 position so that means, 16 x 16 x 4 + 11 x 16 + 5 for the position of A - F we get 1205 (in decimal) 6-digit hexadecimal start with a letter (A-F) and end with a numeral (0-9)

c) there are a total of 2.1875, 3-digit hexadecimal start with a letter (A-F) or end with a numeral (0-9) (or both) as on the left side is "2", that is the whole number part and the 3 is in the "sixteenths" position, meaning "3 sixteenths", or 3/16 so, 2.3 is "2 and 3 sixteenths" (=2.1875 in Decimal)

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The regular expression is (0|1)[0(0|1)1(0|1)] | (0|1)[1(0|1)0(0|1)], which matches any string that is at least 6 symbols long and contains at least one 0 and at least one 1.

One possible regular expression for the language {w | w is at least 6 symbols long and contains at least one 0 and at least one 1} is:

(0|1)[0(0|1)1(0|1)] | (0|1)[1(0|1)0(0|1)]

This regular expression matches any string that satisfies the following conditions:

The string contains at least one 0 and at least one 1.

The string is at least 6 symbols long.

The string can have any number of 0s and 1s before and after the first 0 or 1, but it must contain at least one of each before and after the first 0 or 1.

For example, this regular expression matches strings like "0101010", "1000001", "1110010", but does not match strings like "101", "11111", "0000000".

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Complete question:

Alphabet = {0,1}.

Define a regular expression for the language {w | w is at least 6 symbols long and contains at least one 0 and at least one 1}

In equation 1,  4t + t = 12, so t = 8.

In equation 2, 4f = 8, so f = 2.

Isaiah made two free throws and eight two point shots.

Equations usually contain an equation symbol, such as an equals sign (=) or an inequality symbol, such as a less than sign (<).

We can define the variables as ‘f’ for the number of free throws and ‘t’ for the number of two point shots.

The system of equations for this problem is as follows:

f + t = 12  (Equation 1)

f = 4t  (Equation 2)

Equation 1 states that the total number of points that Isaiah scored is equal to the sum of the number of free throws and the number of two point shots. Equation 2 states that the number of free throws is four times the number of two point shots.

To solve this system of equations, we can use the substitution method. First, substitute 4t for f in equation 1 and solve for t. 4t + t = 12, so t = 8. Then substitute 8 for t in equation 2 and solve for f. 4f = 8, so f = 2.

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In equation 1,  4t + t = 12, so t = 2.4≈3

In equation 2, f = 4t, so f = 9.6≈10

Isaiah made three free throws and ten two point shots.

Equations usually contain an equation symbol, such as an equals sign (=) or an inequality symbol, such as a less than sign (<).

We can define the variables as ‘f’ for the number of free throws and ‘t’ for the number of two point shots.

The system of equations for this problem is as follows:

f + t = 12  (Equation 1)

f = 4t  (Equation 2)

Equation 1 states that the total number of points that Isaiah scored is equal to the sum of the number of free throws and the number of two point shots.

Equation 2 states that the number of free throws is four times the number of two point shots.

To solve this system of equations, we can use the substitution method. First, substitute 4t for f in equation 1 and solve for t.

4t + t = 12

so t = 2.4

Then substitute 2.4 for t in equation 2 and solve for f.

f = 4t

so f = 9.6.

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The length of AC after dilation will be 20. Then the measure of the angle ∠A' will be 37°.

We have given that,

AABC is dilated by a factor of 4 to produce AA'B'C'.

Dilation is the process of increasing the size of an item without affecting its form. Depending on the scale factor, the object's size can be raised or lowered.

The triangle ΔABC is dilated by a factor of 4 to produce the triangle ΔA'B'C'.

Then the length of AC after dilation. We have

A'C' = scale factor x AC

A'C' = 4 x 5

A'C' = 20

Then the measure of the angle ∠A' will be

There is no effect of dilation on the angle.

∠A = ∠A' = 37°

Thus, the correct option is A.

More about the dilation link is given below.

https://brainly.com/question/2856466

Answer:

i think this is the answer!! good luck

Answer:

(h°f)(n) = 32n - 9

Step-by-step explanation:

h(f(n)) = h(-8n+4)

h(-8n+4) = -4(-8n+4) + 7

h(-8n+4) = 32n - 9

(h°f)(n) = 32n - 9

Answer:

125

Step-by-step explanation:

We Take

625 / 5 = 125

So, the answer is 125

G(x) is defined instead of F(x) because the property of memoryless is expressed in terms of G(x). Next, professor refers that there is rational numbers in the set of real numbers, so rational number is dense.  G(x) is an exponential distribution with some rate parameter λ because G(x) has the memoryless property.

The reason why the function G(x) is defined as G(x) = P(X > x) = 1 - F(x) instead of just F(x) is because the memoryless property is expressed in terms of G(x).

Specifically, the memoryless property says G(s+t) = G(s)G(t), which means that the probability of X being greater than s+t is equal to the probability of X being greater than s multiplied by the probability of X being greater than t. This property is easier to work with when expressed in terms of G(x) rather than F(x).

When the professor says that taking the limit of rational numbers gives you real numbers, he is referring to the fact that the set of rational numbers is dense in the set of real numbers. This means that between any two real numbers, there exists a rational number.

In the context of the proof, this means that if G(mn) = G(t)^mn holds for all rational numbers mn, then it also holds for all real numbers x = mn, where mn is the limit of a sequence of rational numbers.

To see why G(x) = G(1)x looks like an exponential function, we can rewrite it as G(x) = e^(ln(G(1))x). Now, suppose we define λ = -ln(G(1)). Then we have G(x) = e^(-λx), which is the probability density function of an exponential distribution with rate parameter λ.

Thus, the assumption that G(x) has the memoryless property implies that G(x) is an exponential distribution with some rate parameter λ.

To know more about an exponential distribution:

https://brainly.com/question/30320669

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