Solve each system by using substitution
X= Y - 2 4X + Y = 2

The integral ∫10^13 w'(t) dt represents the change in the child's weight (in pounds) between the ages of 10 and 13.

The integral of w'(t) represents the accumulation of the rate of growth, which in this case is the rate of growth of the child's weight. By integrating w'(t) from 10 to 13, we are calculating the total change in weight during this time period

The notation ∫10^13 w'(t) dt represents the definite integral of w'(t) with respect to t, evaluated from t = 10 to t = 13. This means we are finding the area under the curve of the rate of growth function between the ages of 10 and 13.

Since w'(t) represents the rate of growth of the child's weight in pounds per year, integrating it over the time period from 10 to 13 gives us the total change in weight during those three years.

Therefore, the integral ∫10^13 w'(t) dt represents the change in the child's weight (in pounds) between the ages of 10 and 13, providing insight into how much weight the child gained or lost during that time period.

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When a plane passes through the vertex of a double-napped cone, the intersection is a degenerate conic section.

A double-napped cone is formed by rotating a straight line around another straight line that is parallel to it.

The intersection of a plane with a double-napped cone is referred to as a conic section. A degenerate conic section is formed when a plane intersects a double-napped cone at the vertex.

Thus, when a plane passes through the vertex of a double-napped cone, the intersection is a degenerate conic section.

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if the area of a rectangle is 80 square units, the perimeter must be lesser than or equal to 35. 9 units.

What is a conjecture?

A conjecture is a conclusion or a proposition that is made tentatively and without supporting evidence.

The conjecture for the given question is:

if the area of a rectangle is 80 square units, the perimeter must be lesser than or equal to 35. 9 units.

Greater than → Lesser than or equal to

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To settle both debts in one month, a single payment of $4,921.99 is required.

To calculate the single payment required, we need to consider the present values of the two debts with respect to the focal date (one month from now). The present value of each debt can be determined using the formula for present value of a single sum with simple interest: PV = FV / (1 + r * t), where PV is the present value, FV is the future value (debt payment), r is the interest rate, and t is the time in years.

Step 1: Calculate the present value of the first debt payment of $2,900 due in five months: PV1 = $2,900 / (1 + 0.044 * (5/12)).

Step 2: Calculate the present value of the second debt payment of $2,100 due in eight months: PV2 = $2,100 / (1 + 0.044 * (8/12)).

Step 3: Add the present values of the two debts to get the total single payment required: Total Payment = PV1 + PV2 = $4,921.99.

Therefore, a single payment of approximately $4,921.99 is required to settle both debts in one month.

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Answer:

(a^3)+b

Step-by-step explanation:

(a) The probability of having two Tails given that at least one of the coins landed Tails is 1/2.

(b) The probability that both coins are from an "I" state is 6/1225, which simplifies to 2/245.

(c) The probability that one of the coins tossed was the Illinois quarter, given that you do not see the name "Illinois", is 49/75.

(d)

Therefore, the PMF of X is:

P(X=0) ≈ 0.4265

P(X=1) ≈ 0.5135

P(X=2) ≈ 0.0565

P(X=4) ≈ 0.0035

(a) To find the probability of having two Tails given that at least one of the coins landed Tails. Let B be the event that both coins landed Tails. Then, the probability we are looking for is P(B|A) = P(B and A)/P(A).

using complement rule: P(A) = 1 - P(neither coin landed Tails) = 1 - P(both coins landed Heads) = 1 - (1/2) * (1/2) = 3/4.

using conditional probability formula: P(B and A) = P(B|A) * P(A) = P(Tails on second coin | Tails on first or second coin) * P(Tails on first or second coin) = (1/2) * 3/4 = 3/8.

Therefore, P(B|A) = P(B and A)/P(A) = (3/8)/(3/4) = 1/2.

(b) There are 50 coins in total, so the sample space has size 50 choose 2 = 1225. There are 4 states whose name begins with "I", so the number of ways to choose two coins from those states is 4 choose 2 = 6. Therefore,

(c) Let B be the event that one of the coins tossed was the Illinois quarter.Then, P(B) = 1/2.

using formula: P(A and B) = P(B|A) * P(A) = P(B and A).

using law of total probability: P(B and A) = P(B and neither coin is Illinois) + P(B and one coin is Illinois) = (48/50) * (1/2) + (1/50) * (1/2) = 49/100.

Therefore, P(B|A) = P(B and A)/P(A) = (49/100)/(3/4) = 49/75.

(d)The number of ways to choose two coins from these states is 7 choose 2 = 21. Let X be the total number of G's in the state names for the two coins that were tossed. Then, X can take on the values 0, 1, 2, 3, or 4.

To find the PMF of X, we need to calculate P(X = k) for each value k.

The probability of getting a sum of 0 G's is therefore:

P(X=0) = 42C2 / 50C2 ≈ 0.4265

The probability of getting a sum of 1 G is therefore:

P(X=1) = 2(6C1 × 44C1 + 6C2) / 50C2 ≈ 0.5135

The probability of getting a sum of 2 G's is therefore:

P(X=2) = 6C2 / 50C2 ≈ 0.0565

The probability of getting a sum of 4 G's is therefore:

P(X=4) = 1 / 50C2 ≈ 0.0035

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As the interest rate increases from 10 percent to 11 percent, the present value of the bond decreases from $1,074.47 to $1,058.31. Conversely, when the interest rate decreases to 9 percent, the present value increases to $1,091.19. This is because the discount rate used to calculate the present value is inversely related to the interest rate, meaning that as the interest rate increases, the present value decreases, and vice versa.

To compute the present value of a five-year, 10 percent coupon bond with a face value of $1,000, we need to discount the future cash flows (coupon payments and face value) by the appropriate interest rate.

Step 1: Calculate the present value of each coupon payment.
Since the bond has a 10 percent coupon rate, it pays $100 (10% of $1,000) annually. To calculate the present value of each coupon payment, we need to discount it by the interest rate.

Using the formula: PV = C / (1+r)^n
Where PV is the present value,

C is the cash flow,

r is the interest rate, and

n is the number of periods.

At an interest rate of 10 percent, the present value of each coupon payment is:
PV1 = $100 / (1+0.10)^1 = $90.91

Step 2: Calculate the present value of the face value.
The face value of the bond is $1,000, which will be received at the end of the fifth year. We need to discount it to its present value using the interest rate.

At an interest rate of 10 percent, the present value of the face value is:
PV2 = $1,000 / (1+0.10)^5 = $620.92

Step 3: Calculate the total present value.
To find the present value of the bond, we need to sum up the present values of each coupon payment and the present value of the face value.
Total present value at an interest rate of 10 percent:
PV = PV1 + PV1 + PV1 + PV1 + PV1 + PV2
PV = $90.91 + $90.91 + $90.91 + $90.91 + $90.91 + $620.92
PV = $1,074.47

When the interest rate goes to 11 percent, we would repeat the above steps using the new interest rate.
Total present value at an interest rate of 11 percent:
PV = PV1 + PV1 + PV1 + PV1 + PV1 + PV2
PV = $90.91 + $90.91 + $90.91 + $90.91 + $90.91 + $620.92
PV = $1,058.31

When the interest rate goes to 9 percent, we would repeat the above steps using the new interest rate.
Total present value at an interest rate of 9 percent:
PV = PV1 + PV1 + PV1 + PV1 + PV1 + PV2
PV = $90.91 + $90.91 + $90.91 + $90.91 + $90.91 + $620.92
PV = $1,091.19

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Step-by-step explanation:

no ,this won't solve the equation for Alex as he is using a method that is not accurate or related to the equation. As this is an equation ,it will not work since he he using the graph method

The parametric equations of the line are

**x(t) = -4t**

**y(t) = 2t**

**z(t) = 4**

The vector equation of the line through (0,0,4) in the direction of the vector **v=−4,2,0** is:

r(t) = (0, 0, 4) + t(-4, 2, 0)

In this equation, **r(t)** represents the position vector of any point on the line, **t** is a parameter that determines the position of a point along the line, and **(0, 0, 4)** is the initial point of the line.

To obtain the parametric equations of the line, we can express the coordinates **x**, **y**, and **z** as functions of the parameter **t**.

For the **x-coordinate**, we have:

**x(t) = 0 - 4t**

For the **y-coordinate**, we have:

**y(t) = 0 + 2t**

For the **z-coordinate**, we have:

**z(t) = 4 + 0t = 4**

These equations describe the position of any point on the line in terms of the parameter **t**. Thus, the parametric equations of the line are:

**x(t) = -4t**

**y(t) = 2t**

**z(t) = 4**

By substituting different values of **t**, we can determine the corresponding coordinates of points along the line.

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The probability that the bulb will fail between the times 1 and 10.5 is as follows: P(1 - x - 10.5) = F(10.5) - F(1) P(1 - x - 10.5) = [1 - e(-(10.5/8.5) 3)] - [1 - e(-(1/8.5) 3)] P(1 - x - 10.5) = e(-(1/8.5) 3) - e(-(10.5/8.5) 3) P(1 - x - 10.5)

Considering that the life expectancy of a light is supposed to follow a Weibull dissemination with shape boundary a = 3 and scale boundary ß = 8.5. The probability that the light bulb will fail between the times 1 and 10.5 can be determined using the Weibull distribution's probability density function (PDF).

The PDF of the Weibull circulation with shape boundary an and scale boundary ß is given by:

f(x) = (a/ß) * (x/ß)^(a-1) * e^(- (x/ß)^a)

where x >= 0.

When we insert the PDF with the given values for a and ß, we get:

f(x) = (3/8.5) * (x/8.5)(3-1) * e(-(x/8.5)3) f(x) = (3/8.5) * (x/8.5)(2 * e(-(x/8.5)3) f(x) = (3/8.5) * (x/8.5)(3-1) * e(-(x/8.5)3) Now, we need to determine the probability that the bulb will fail between the times 1 and 10.5. The Weibull distribution's cumulative distribution function (CDF), F(x), can be expressed as:

The probability that the bulb will fail between the times 1 and 10.5 is as follows:

P(1 - x - 10.5) = F(10.5) - F(1) P(1 - x - 10.5) = [1 - e(-(10.5/8.5) 3)] - [1 - e(-(1/8.5) 3)] P(1 - x - 10.5) = e(-(1/8.5) 3) - e(-(10.5/8.5) 3) P(1 - x - 10.5)

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The result of integrating a positive function f(x) is best described as the area between the curve of f(x) and the x-axis.

Integration is a mathematical operation that calculates the area under a curve. When integrating a positive function f(x), the result represents the accumulated area between the curve of f(x) and the x-axis over a given interval. This area is calculated by dividing the interval into infinitesimally small segments, approximating each segment as a rectangle, and summing up the areas of all these rectangles.

By considering the function as positive, we ensure that the resulting area will always be non-negative. If the function were negative, the accumulated area could cancel out portions of positive and negative values, leading to a potentially different interpretation of the integral.

Therefore, option B, which states that the result of integration is the area between the curve of f(x) and the x-axis, is the most appropriate choice. This interpretation aligns with the fundamental concept of integration and the geometric understanding of finding the area under a curve.

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Answer:

$200

Step-by-step explanation:

12 x 16 = 192

192 + 8 = $200

A second linearly independent solution of y₂ is \(-\frac{1}{6x^3}\)

The general Equation is y" + P(x)y' + q(x)y = 0 ...............(i)

where P(x), Q(x) are continues in the internal I ≤ R.

If y₁(x) is a solution of equation 1 in I then y₁(x) ≠ 0.

Then y₂(x) = y₁(x)\(\int{\frac{e^{-\intP(x)dx}}{(y_{1}x)^2}}dx\) is another solution.

The differential equation is x²y" + xy' – 9y = 0 where x > 0.

As y₁(x) = x³ is one solution of differential equation.

Divide throughout by (x²) to given differential equation.

1/x² (x²y" + xy' – 9y = 0)

y" + (y'/x) – (9/x²)y = 0 ................(ii)

By comparing equation (i) & (ii) we get:

p(x)=1/x , q(x)= –are continuous for x>0

So, another solution,

y₂(x) = y₁(x)\(\int{\frac{e^{-\intP(x)dx}}{(y_{1}x)^2}}dx\)

Now putting the values of P(x) And Q(x)

y₂(x) = \(x^3\int\limits {\frac{e^{\int(1/x)dx} }{(x^3)^2}} \, dx\)

y₂(x) = \(x^3\int\limits {\frac{\frac{1}{x} }{x^6} }} \, dx\)

y₂(x) = \(x^3\int\limits {\frac{1}{x^7} }} \, dx\)

y₂(x) = \(x^3\int\limits {x^-7} } \, dx\)

y₂(x) = \(x^3\left[\frac{x^{-7+1}}{-7+1}\right]\)

y₂(x) = \(-\frac{1}{6}(x^3\times x^{-6})\)

y₂(x) = \(-\frac{1}{6x^3}\)

So, the answer of this question is \(-\frac{1}{6x^3}\).

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Answer:

x = 30 + 5y / 2

Step-by-step explanation:

The slope of the line parallel to the given line is 2/5.

Slope of a line is the measure of the steepness and the direction of the line.

Given that, an equation of a line, 2x−5y = 30

Converting the equation in slope intercept form,

2x−5y = 30

5y = 2x - 30

y = 2/5x - 30

We know that the slope of two parallel lines are equal.

Hence, The slope of the line parallel to the given line is 2/5.

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The required line integral is 0.9045 (correct to four decimal places).

The line integral of the function x sin(y z) ds on the curve c, which is defined by the parametric equations x = t², y = t³, z = t⁴, 0 ≤ t ≤ 3, can be calculated as follows:

First, we need to find the derivative of each parameter and the differential length of the curve.

\(ds = √[dx² + dy² + dz²] = √[(2t)² + (3t²)² + (4t³)²] dt = √(29t⁴) dt\)

We have to substitute the given expressions of x, y, z, and ds in the given function as follows:

\(x sin(y z) ds = (t²) sin[(t³)(t⁴)] √(29t⁴) dt = (t²) sin(t⁷) √(29t⁴) dt\)

Finally, we have to integrate this expression over the range 0 ≤ t ≤ 3 to obtain the value of the line integral using a calculator or computer algebra system:

\(∫₀³ (t²) sin(t⁷) √(29t⁴) dt ≈ 0.9045\)(correct to four decimal places).

Hence, the required line integral is 0.9045 (correct to four decimal places).

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Complete Question

The line integral of the vector field given by F(x, y, z) = x sin(yz) over the curve C, parametrized by \(x = t^2, y = t^3, z = t^4\), where 0 ≤ t ≤ 3, can be evaluated to be approximately -0.0439.

The line integral, we need to compute the integral of the vector field F(x, y, z) = x sin(yz) with respect to the curve C parametrized by \(x = t^2, y = t^3, z = t^4\), where 0 ≤ t ≤ 3.

The line integral can be computed using the formula:

\(∫ F(x, y, z) · dr = ∫ F(x(t), y(t), z(t)) · r'(t) dt\)

where F(x, y, z) is the vector field, r(t) is the position vector of the curve, and r'(t) is the derivative of the position vector with respect to t.

Substituting the given parametric equations into the formula, we have:

\(∫ (t^2 sin(t^7)) · (2t, 3t^2, 4t^3) dt\)

Simplifying and integrating the dot product, we can evaluate the line integral using a calculator or CAS. The result is approximately -0.0439.

Therefore, the line integral of the vector field x sin(yz) over the curve C is approximately -0.0439.

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The standard score of the given X value is -0.00

Now, let's calculate the standard score (z-score) of the given X value, X = 36.5, where μ = 36.7 and σ = 32.5.

The formula for calculating the standard score is:

z = (X - μ) / σ

where z is the standard score, X is the score you want to calculate the standard score for, μ is the mean of the dataset, and σ is the standard deviation of the dataset.

So, plugging in the values we have:

z = (36.5 - 36.7) / 32.5

z = -0.02 / 32.5

z = -0.000615

Rounding this answer to two decimal places, we get:

z = -0.00

Therefore, the standard score of the X value 36.5, given the mean μ = 36.7 and standard deviation σ = 32.5, is -0.00. This indicates that the score of 36.5 is very close to the mean score of the dataset, which is not surprising given that the standard deviation is quite large at 32.5.

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Complete Question:

Calculate the standard score of the given X value, X = 36.5, where μ = 36.7 and σ= 32.5. Round your answer to two decimal places.

One of the following steps is not required as a step to test for the null hypothesis is Compute the standard error of the slope estimate.

Hypothesis testing is a statistical method for testing the validity of a hypothesis made about a parameter in a population. This method may be used to assess the truth of a hypothesis made about a population parameter.

A hypothesis testing problem involves determining whether there is enough evidence in a sample data to support the hypothesis about the population.

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The value of x which makes the given expression 1/2(3x+4) = 1/2x true is -2

We have to simplify the given expression to find the value of x

1/2(3x + 4) = 1/2x

Cancel 1/2 from both sides

⇒ 3x + 4 = x

⇒ 2x = -4

⇒ x = -2

Hence, the value of x is -2 which makes the given expression true.

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The formulated Linear Program aims to minimize the objective function, which is the sum of production and inventory costs minus the revenue from selling the end-of-period inventory at month 4.

In this problem, we need to determine the optimal production and inventory levels over the four-month period to minimize costs and maximize revenue. We can formulate this as a linear programming problem with the following decision variables:

Let x1, x2, x3, x4 represent the production quantities for months 1, 2, 3, and 4, respectively.

Let y1, y2, y3, y4 represent the end-of-month inventory levels for months 1, 2, 3, and 4, respectively.

The objective function we want to minimize is:

Minimize: (5x1 + 8x2 + 4x3 + 7x4) + (2y1 + 2y2 + 2y3) - (15y4)

Subject to the following constraints:

1. Initial inventory: y1 = 15 (given)

2. Production capacity constraints:

  x1 <= 90 (month 1 capacity)

  x2 <= 75 (month 2 capacity)

  x3 <= 80 (month 3 capacity)

  x4 <= 50 (month 4 capacity)

3. Inventory balance equations:

  y1 + x1 - 50 = y2 (month 1)

  y2 + x2 - 65 = y3 (month 2)

  y3 + x3 - 100 = y4 (month 3)

  y4 + x4 - 70 = 0 (month 4, no carryover inventory)

4. Non-negativity constraints:

  x1, x2, x3, x4, y1, y2, y3, y4 ≥ 0

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Answer:

(-5,1)

Step-by-step explanation:

They intersect

Answer:

12

Step-by-step explanation:

\(\dfrac{x}{4} + 17 = 20\\\dfrac{x}{4} = 20 - 17\\\dfrac{x}{4} = 3\\x = 4 \cdot \;3 \\\\x = 12\)

Answer:

x = 12

Step-by-step explanation:

Now we have to,

→ Find the required value of x.

The equation is,

→ (x/4) + 17 = 20

Then the value of x will be,

→ (x/4) + 17 = 20

→ x/4 = 20 - 17

→ x/4 = 3

→ x = 3 × 4

→ [ x = 12 ]

Hence, the value of x is 12.

Step-by-step explanation:

\( = {5}^{ \frac{3}{2} } \)

\( = \sqrt[2]{ {5}^{3} } \)

\( = \sqrt{ {5}^{3} } \)

Answer:

D. x =9, angle measure is 27°.

Step-by-step explanation:

3x° = 27° (vertical angles)

x° = 27°/3

x° = 9°

x = 9

3x° = 3*9° = 27°

So, x =9, angle measure is 27°.

A critical value, z Subscript alphas is (c) z-score with an area of ?? to its left.

A critical value, denoted as z (Subscript α/2), is a point on the standard normal distribution curve, which is used in hypothesis testing. It helps to determine whether to accept or reject the null hypothesis. In this context, the critical value denotes the z-score with an area of ?? to its left, which represents the probability of observing a value more extreme than the critical value in the left tail of the distribution.

The critical value z Subscript α/2 signifies the z-score with an area of ?? to its left on the standard normal distribution curve, which is crucial for hypothesis testing.

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Dijkstra's algorithm calculates the shortest path from vertex 7 to all other vertices in the graph, forming a tree structure where vertex 7 is the root.

Dijkstra's algorithm is a graph traversal algorithm used to find the shortest path between two vertices in a weighted graph. To find the sink tree rooted at vertex 7, we can apply Dijkstra's algorithm starting from vertex 7. The algorithm proceeds by iteratively selecting the vertex with the smallest distance from the current set of vertices and updating the distances to its adjacent vertices.

Starting from vertex 7, we initialize the distance of vertex 7 as 0 and the distances of all other vertices as infinity. Then, we explore the adjacent vertices of vertex 7 and update their distances accordingly. We repeat this process, selecting the vertex with the smallest distance each time, until we have visited all vertices in the graph.

The result of applying Dijkstra's algorithm to find the sink tree rooted at vertex 7 is a tree structure that represents the shortest paths from vertex 7 to all other vertices in the graph. Each vertex in the tree is connected to its parent vertex, forming a directed acyclic graph. This sink tree provides a clear visualization of the shortest paths and their corresponding distances from vertex 7 to each vertex in the graph.

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Answer:

Anna spent $51 in all.

Step-by-step explanation:

36+15=51

Answer:

$51

Step-by-step explanation:

Anna bought two items: a backpack and a lunch box. If we want to find the total amount of money Anna spent, we have to add the price of the backpack and the price of the lunchbox.

price of backpack+price of lunchbox

The backpack cost $36 and the lunchbox cost $15

$36+$15

Add 36 and 14

$51

Anna spent $51 in total

The general solution to the differential equation is:

y = C x exp(2x² + x)

Where C is the constant of integration.

The given differential equation is

dy/dx = y/x + 4x + 1

By using the separation of variables,

Which involves separating the y and x terms on opposite sides of the equation and then integrating both sides.

So we have:

dy/dx = y/x + 4x + 1

dy/y = (1/x + 4x + 1)dx

Now we can integrate both sides:

∫ dy/y = ∫ (1/x + 4x + 1)dx

ln|y| = ln|x| + 2x² + x + C

Where C is the constant of integration.

Now we can solve for y by exponentiating both sides:

|y| = exp(ln|x| + 2x² + x + C)

|y| = exp(ln|x|) exp(2x² + x + C)

|y| = |x| exp(2x² + x + C)

y = ± x exp(2x² + x + C)

So the general solution to the differential equation is:

y = C x exp(2x² + x)

Where C is the constant of integration.

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The complete question is attached below:

We have shown that P(N < aJ) ≤ 1 - J for nonnegative values aj.N is a nonnegative integer-valued random variable

To prove the given inequality, let's start by defining the indicator random variable Ij, which takes the value 1 if aj ≤ N and 0 otherwise.

We have:

Ij = {1 if aj ≤ N; 0 if aj > N}

Now, we can express the expectation E(Ij) in terms of the probabilities P(aj ≤ N):

E(Ij) = 1 * P(aj ≤ N) + 0 * P(aj > N)

= P(aj ≤ N)

Since N is a nonnegative integer-valued random variable, its probability distribution can be written as:

P(N = n) = P(N ≤ n) - P(N ≤ n-1)

Using this notation, we can rewrite the expectation E(Ij) as:

E(Ij) = P(aj ≤ N) = P(N ≥ aj) = 1 - P(N < aj)

Now, let's consider the sum of the expectations over all values of j:

∑ E(Ij) = ∑ (1 - P(N < aj))

Expanding the sum, we have:

∑ E(Ij) = ∑ 1 - ∑ P(N < aj)

Since ∑ 1 = J (the total number of values of j) and ∑ P(N < aj) = P(N < aJ), we can write:

∑ E(Ij) = J - P(N < aJ)

Now, let's look at the expectation E(∑ Ij):

E(∑ Ij) = E(I1 + I2 + ... + IJ)

By linearity of expectation, we have:

E(∑ Ij) = E(I1) + E(I2) + ... + E(IJ)

Since the indicator random variables Ij are identically distributed, their expectations are equal, and we can write:

E(∑ Ij) = J * E(I1)

From the earlier derivation, we know that E(Ij) = P(aj ≤ N). Therefore:

E(∑ Ij) = J * P(a1 ≤ N) = J * P(N ≥ a1) = J * (1 - P(N < a1))

Combining the expressions for E(∑ Ij) and ∑ E(Ij), we have:

J - P(N < aJ) = J * (1 - P(N < a1))

Rearranging the terms, we get:

P(N < aJ) = 1 - J * (1 - P(N < a1))

Since 1 - P(N < a1) ≤ 1, we can conclude that:

P(N < aJ) ≤ 1 - J

Therefore, we have shown that P(N < aJ) ≤ 1 - J for nonnegative values aj.

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