let be a geometric sequence with and ratio . for how many is it true that the smallest such that is ?

The function g(t) is defined as:

g(t) = 1/√(16-t^2)

The function is continuous for all values of t that satisfy the following conditions:

The denominator is non-zero:

The denominator of the function is √(16-t^2). Therefore, the function is undefined when 16-t^2 < 0, or when t is outside the interval [-4,4].

There are no vertical asymptotes:

The function does not have any vertical asymptotes, because the denominator is always positive.

Thus, the function g(t) is continuous on the interval [-4,4].

Answer:

Let's start by expressing "Z increased by 16%" using algebra.

Let Z be the original value of some quantity.

To increase Z by 16%, we need to add 16% of Z to Z:

Z + 0.16Z

Simplifying this expression by factoring out Z, we get:

Z(1 + 0.16)

Combining like terms, we have:

Z(1.16)

Therefore, "Z increased by 16%" can be expressed algebraically as:

Z increased by 16% = Z(1.16)

Answer:

z(1.16)

Step-by-step explanation:

Part A. The probability pf drawing a purple card out of the hat is 28%.

Part B. The experimental probability is 2% greater than the theoretical probability.

The probability that a specific event will occur is known as probability. The ratio of favourable outcomes to all other possible outcomes serves as a stand-in for the likelihood that an event will occur.

In numerous disciplines, including mathematics, statistics, physics, economics, and computer science, uncertain events are described and understood using probability theory. It is used to analyse risks, make decisions, and forecast events.

Now in the given question,

Total cards in the hat = 12 + 17 + 14 + 7 = 50 cards

Total purple cards in the hat = 14

Probability of getting a purple card from the hat = 14/50

= 0.28

= 28%

Now similarly for the experiment,

Probability = 324/1080

= 0.3

= 30%

Therefore, the experimental probability is 30% - 28% = 2% greater than the theoretical probability.

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a) The domain of the function is  [0, 7.1875], while the range is [0,23]. Considering the domain and the range, the graph of the function is given by the image presented at the end of the answer.

b) The trip had a duration of 5.1875 hours.

The function for this problem is defined as follows:

g(t) = 23 - 3.2t.

The domain is the set of input values that can be assumed by the function. The time cannot have negative measures, hence the lower bound of the domain is of zero, while the gas cannot be negative, hence the upper bound of the domain is given as follows:

23 - 3.2t = 0

3.2t = 23

t = 23/3.2

t = 7.1875 hours.

The range is given by the set of all output values assumed the function, which are the values of the gas, hence it is [0,23].

The graph is a linear function between points (0, 23) and (7.1875, 0).

At the end of the trip there were 6.4 gallons left, hence the length of the trip is obtained as follows:

23 - 3.2t = 6.4

t = (23 - 6.4)/3.2

t = 5.1875 hours.

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When a shape is dilated, the size of the shape changes. The true statement is (d) The scale factor is 2.5.

Dilation:

Dilation is the process of changing the size of an object or shape by reducing or increasing its size by a specific scale factor. For example, a circle with a radius of 10 units shrinks to a circle with a radius of 5 units. Applications of this method are in photography, arts and crafts, sign making and more.

According to the Question:

How to determine the scale factor

In figure A, we have:

Length = 0.6

In figure B, we have:

Length =1.5

The scale factor is then calculated as:

K = 1.5/0.6

Dividing the equation:

k = 2.5

Hence, the true statement is (d) The scale factor is 2.5.

Complete Question:

The first figure is dilated to form the second figure. Which statement is true?

The scale factor is 0.4.

The scale factor is 0.9.

The scale factor is 2.1.

The scale factor is 2.5.

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Therefore, the volume of the sphere to the nearest hundredth is 724,775.70 cubic millimeters.

Volume is a measurement of the amount of space occupied by a three-dimensional object. It is often expressed in units such as cubic meters (m³), cubic centimeters (cm³), cubic feet (ft³), or gallons (gal), depending on the context. The volume of a solid object can be calculated by multiplying its length, width, and height or using a specific formula depending on the shape of the object. For example, the volume of a rectangular box can be calculated as length x width x height, while the volume of a cylinder can be calculated as π x radius² x height. In general, volume is an important concept in many fields, including physics, chemistry, engineering, and architecture. It is often used to describe the capacity of containers, the displacement of fluids, and the amount of material used in construction or manufacturing.

Here,

The formula for the volume of a sphere is given as V = (4/3)πr³, where r is the radius of the sphere and π is approximately 3.14.

Substituting the given value of the radius, we get:

V = (4/3) x 3.14 x 48³

V ≈ 724,775.68 cubic millimeters

Rounding this value to the nearest hundredth, we get:

V ≈ 724,775.68 ≈ 724,775.70 cubic millimeters (rounded to two decimal places)

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The probability of picking a vowel, replacing it, and then picking a consonant from the word "SLEEP" is approximately 0.096 or 9.6%.

Probability is usually expressed as a number between 0 and 1, with 0 meaning that the event is impossible and 1 meaning that the event is certain.

There are two vowels (E) and three consonants (S, L, P) in the word "SLEEP".

The probability of picking a vowel on the first draw is 2/5, because there are two vowels out of five letters total.

Since we replace the vowel we picked, the probability of picking another vowel on the second draw is also 2/5.

The probability of picking a consonant on the third draw is 3/5, because there are three consonants left out of five letters total.

Therefore, the probability of picking a vowel, replacing it, and then picking a consonant from the word "SLEEP" is:

(2/5) x (2/5) x (3/5) = 12/125 or approximately 0.096 or 9.6%.

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The triangle's other leg, side B, measures 12 cm in length.

When one of the interior angles is 90 degrees, or a right angle, the triangle is said to be a right triangle. The three internal angles of a triangle add up to 180 degrees in a right triangle because one angle must always be 90 degrees and the other two must always total to 90 degrees (they are complementary).

We can observe that the given triangle is a right triangle because angle A's measure is 90 degrees. The hypotenuse, which is represented by the letter "c," is the side that is opposite the right angle. The legs are the other two sides, and they are indicated by "a" and "b".

We are told that the hypotenuse (side c) is 13 cm long and that one leg (side a) is 5 cm long. The length of the other leg must be determined (side b).

The Pythagorean theorem, which asserts that in a right triangle, can be used.

a² + b² = c²

Inputting the values provided yields:

5² + b² = 13²

25 + b² = 169

b² = 144

b = 12

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The equation can be rearranged to the form y = -qx + r. This is the equation of a straight line, which can be graphed. The point of intersection of the two lines, ak + bl = 0 and y = -qx + r, is the solution for the two variables (k and l).

The equation ak + bl = 0 is a linear equation in two variables and is solved using the method of elimination. The equation can be written in the form ax + by = c, where a, b, c are constants. To solve this equation, both sides of the equation should be divided by the coefficient of one of the variables (a or b). This will result in a equation of the form x + qy = r, where q and r are constants. Then, the equation can be rearranged to the form y = -qx + r. This is the equation of a straight line, which can be graphed. The point of intersection of the two lines, ak + bl = 0 and y = -qx + r, is the solution for the two variables (k and l). The two variables can then be calculated using the point of intersection by substituting the x and y values into the two equations. In this way, the two variables k and l can be found such that ak + bl = 0.

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What are the integers k and l such that ak + bl = 0?

Answer:

x = 4

Step-by-step explanation:

using the rule of exponents

• [tex]a^{-m}[/tex] = [tex]\frac{1}{a^{m} }[/tex] , then

[tex]4^{-x}[/tex] = [tex]\frac{1}{4^{x} }[/tex]

and 256 = [tex]4^{4}[/tex]

then

[tex]\frac{1}{4^{x} }[/tex] = [tex]\frac{1}{4^{4} }[/tex]

so

[tex]4^{x}[/tex] = [tex]4^{4}[/tex]

since bases on both sides are equal, bot 4 then equate exponents

x = 4

B, No, they are not independent events because the probability of a student preferring logo B (0.78) is different from the overall probability of preferring logo B (0.89), which includes both students and teachers.

To determine whether being a student and preferring logo B are independent events, we need to compare the probability of a student preferring logo B (P(student logo B)) with the overall probability of preferring logo B (P(logo B)).

P(student logo B) = 0.78 (from the table)

P(logo B) = (86 + 11) / 125 = 0.89

If the two probabilities are equal, then the events are independent. However, in this case, P(student logo B) is not equal to P(logo B), indicating that being a student and preferring logo B are dependent events. Therefore, being a student and preferring logo B are dependent events.

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When x and y are independent variables with x having uniform distribution (0,1) and y having exponential distribution respectively, then e (X+Y) is 3/2, E(XY) is 1/2, E (x-y)2 is 4/3, and e (x2 e24) is -1/3 22etyx u (0,1) > Independent y exp (1) > Independent

The exponential distribution is a continuous probability distribution used in probability theory and statistics that frequently addresses the amount of time until a given event occurs. Events occur continually, independently, and at a steady average pace during this process.

E (x) = 0+1/2 = 1/2

v (x) = [1 – 0]2 / 12 = 1/12

E x2 = v (x) + [Ex]2

      = 1/12 + 1/4 = 1+3/12 = 4/12 = 1/3

E (y) = 1  v(y)=1  

E y2= 1+1= 2

Eety = ∫0etye-ydy  

Eety  =  ∫0e-y (1 – t)dy  

a : E (x+y) = E (x) + E(y)

      E (x+y) = 1/2+1 = 3/2

b: E (xy) = E (x) E(y) = 1/2 x 1 = 1/2

c: E (x+y)2 = E (x2 - 2xy+ y2)

 =  1/3 - 2 x 1/2 + 2

  = 1/3 + 1 = 4/3

d. E x2e24

= 1/3 x -1 = - 1/3

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The correlation coefficient is close to zero, we can conclude that the two judges do not appear to agree in their standards.

Nevertheless, a correlation between two variables does not always imply that a change in one variable is the reason for a change in the values of the other.

There is a causal link between the two occurrences, which means that causation shows that one event is the outcome of the occurrence of the other event. This concept is also known as cause and effect.

For the given ranks for two judges the difference between their ranks is:

d: 0.0 4.0 -2.0 1.0 -0.5 1.5 -1.0 -1.0 0.5 2.0

Squaring the given distance we have:

d²: 0.0 16.0 4.0 1.0 0.25 2.25 1.0 1.0 0.25 4.0

Σd² = 29.75

The Spearman's rank correlation coefficient is given as:

ρ = 1 - (6Σd²)/(n(n²-1))

ρ = 1 - (629.75)/(10(10²-1))

ρ ≈ 0.03

Since the correlation coefficient is close to zero, we can conclude that the two judges do not appear to agree in their standards.

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The complete question is:

The company can produce approximately 425 widgets for $2127.

What is cost function ?

The key concept used here is the concept of cost functions, which is an important concept in economics and business. A cost function is a mathematical function that expresses the total cost of production as a function of the level of output produced. In this case, the cost function is a linear function of the form C = a + bx, where C is the total cost, a is the fixed cost, b is the variable cost per unit, and x is the level of output.

Finding the number of widgets the company can produce given a fixed cost and a variable cost per widget :

To solve this problem, we can set up an equation that relates the total cost to the number of widgets produced.

Let x be the number of widgets produced.

The total cost C is given by:

C = fixed cost + variable cost

C = 1277 + 1.93x

We want to find the number of widgets produced for a total cost of $2127. So we can set up an equation:

2127 = 1277 + 1.93x

Subtracting 1277 from both sides gives:

850 = 1.93x

Dividing both sides by 1.93 gives:

x ≈ 439.9

Since we can't produce a fractional number of widgets, we need to round down to the nearest integer. Therefore, the company can produce approximately 425 widgets for $2127.

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The value of x is 5

Triangles with the same shape but different sizes are said to be similar triangles. To be more specific, two triangles are comparable if their respective sides are proportionate and their corresponding angles are congruent.

Two triangles are similar if corresponding angles are congruent and corresponding sides are proportional.

from the below figure, both the triangles are similar, ∆ABC ≈ ∆EFB

By using Thales's theorem, the ratio of the sides of triangles are;

BE/EA = BF/FC

15/30 = x/10    

x = 5

Therefore the value of x is 5

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The ratio of triangle sides can be calculated using Thales' theory, and it is the value of x is 5

Similar triangles are those with the same shape but varying sizes. To be more precise, two triangles are comparable if their matching angles and respective sides are congruent.

If matching sides are proportional and corresponding angles are congruent, two triangles are similar.

Both triangles in the following figure are comparable ∆ABC ≈ ∆EFB

The ratio of triangle sides can be calculated using Thales' theory, and it is;

BE/EA = BF/FC

15/30 = x/10    

x = 5

Therefore the value of x is 5

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The population distribution of scores has a similar average as the distribution of means.

In a sample distribution of means, the distribution will roughly resemble a normal distribution, the sampling distribution's mean will be equal to the population's mean, and the sampling distribution's standard deviation is based on the Central Limit Theorem.

The central limit theorem (CLT) of probability theory states that the distribution of a sample variable tends towards a normal distribution (i.e., a "bell curve") as the sample size increases, under the premise that all samples are of equal size and regardless of the population's actual distribution shape.

In other terms, the central limit theorem (CLT) is a statistical presumption that the mean of all sampled variables from the same population will be substantially equal to the mean of the entire population, given a large enough sample size from a population with little volatility. These samples likewise resemble a normal distribution in accordance with the law of large numbers, with their variances nearly matching the variance of the population as the sample size rises.

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the actual question is :

When comparing the size of the standard error of the mean with the size of the standard deviation of the underlying distribution of individual scores:

a. the standard error of the mean is always larger

b. the standard error of the mean is always smaller

c. the standard error of the mean is sometimes larger and sometimes smaller, depending on the sample size

d. none of these

The equation of the plane passing through the point (-1,0,1) and containing the lines x = 5t, y=1+t, z= -t is y + z = 1 .

We substitute , t=0 in the given line equations ,

we get; x=0 , y = 1 and z = 0 ;

So, the plane contain the line , Thus plane will also pass through (0,1,0);

Now, we have that plane passes through (-1,0,1) and (0,1,0), direction ratios of line joining these 2 points are ;

⇒ DR₁ = (-1-0 , 0-1 , 1-0) = (-1,-1,1);

So , the line can be written as x/5 = (y-1)/1 = z/-1 = t;

So, the direction ratio of this line will be :

⇒ DR₂ = (5 , 1 , -1);

The Direction Ratio of normal to the plane is = DR₁ × DR₂;

= (-1,-1,1) × (5,1,-1);

= [tex]\left|\begin{array}{ccc}i&j&k\\-1&-1&1\\5&1&-1\end{array}\right|[/tex]

On solving ,

We get;
= 4j + 4k = (0,4,4) ;

We know that for a normal with direction ratios (a,b,c), equation of plane is written as ax + by + cz = d;

We got direction ratio for plane normal = (0,4,4);

So, equation of plane is 0x+ 4y + 4z = d;

the plane passes through the point (-1,0,1) ;

⇒ 4(0) + 4(1) = d ⇒ d = 4;

we get the equation of plane is 4y + 4z = 4;

⇒ y + z = 1.

Therefore, the equation of the required plane is  y + z = 1.

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Answer:

g(x) = 1/2|x - 1| + 2

Step-by-step explanation:

You were almost there, you just got the slope wrong.  

original vertex (h, k): (0, 0)

transformed vertex (h', k'): (1, 2)

original slope:  1

transformed slope: m = (3-2)/(3-1) = 1/2    pick any 2 points to find the slope

equation for the transformed function shown in the graph:

g(x) = 1/2|x - h'| + k'

g(x) = 1/2|x - 1| + 2

The area of the composite shape using the area formula for the different shapes is 460.48ft².

The area of composite shapes refers to the space occupied by any composite shape. A composite shape is a shape that is made by connecting a few polygons to form the required shape.

These figures or shapes can be built from a wide range of shapes, such as triangles, squares, quadrilaterals, etc. Divide a composite item into basic forms such a square, triangle, rectangle, or hexagon to get its area.

Now in the question,

First let us find the area of the semi-circle.

Area of semi-circle = πr²/2

= [3.14 × (16/2) ²]/2

= (3.14 × 8²)/2

= 200.96/2

= 100.48ft²

Now coming to the rectangle,

area of the rectangle = l × b

= 20 × 15

= 300ft²

Now for calculating the area of the triangle,

area = 1/2 × b × h

= 1/2 × 12 × 10

= 60ft²

Therefore, the area of the total figure = 100.48 + 300 + 60 = 460.48ft².

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The probability of obtaining a reading between 0.59°C and 0.88°C is 0.7224 and 0.8106.

The sum of all possible values, weighted by the chance of each value, is equal to the mean of a discrete probability distribution of the random variable X. Each possible number of X must be multiplied by its probability P(x) before being added as a whole to determine the mean. In statistics, the mean is one measure of central trend in addition to the mode and median. The mean is simply the average of the numbers in the specified collection. It suggests that values in a specific data gathering are evenly distributed. In order to find the mean, the total values given in a datasheet must be added, and the result must be divided by the total number of values.

In this question, using the formula,

z-score = (x – μ) / σ

where:

x: individual data value

μ: population mean

σ: population standard deviation

for x=0.59

μ= 0

σ= 1

z-score= 0.59

Probability=0.7224

for x=0.88

z-score= 0.88

Probability=0.8106

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Given that (1, 2, 3] System{1, 4, 7,6] for a system known to be LTI, the impulse response of the system: h[n] = (1/2)*δ[n] + δ[n-1] + (3/2)*δ[n-2]

To compute the impulse response h[n] of a linear time-invariant (LTI) system given its input-output relationship, we can use the convolution sum:

y[n] = x[n] * h[n]

y[n] = (1/2)*(x[n] + 2x[n-1] + 3x[n-2])

y[n] = (1/2)*(δ[n] + 2δ[n-1] + 3δ[n-2])

y[n] = (1/2)*δ[n] + δ[n-1] + (3/2)*δ[n-2]

Thus, the impulse response of the system is:

h[n] = (1/2)*δ[n] + δ[n-1] + (3/2)*δ[n-2],where δ[n] is the impulse signal.

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A) Relative frequency is number of times an event happened over total number of events:

Answer is 21/50 = 0.42

B) On a 6 sides die, there are 3 even numbers and 3 odd numbers, so the theoretical probability of landing on odd would be 3/6 = 0.50

C) Because the die has an equal amount of chance landing on even or odd, both are 3/6, then the dice is not biased.

Samir's new balance is $5,044.17.

To calculate Samir's new balance, add the previous balance, subtract the payment, add the new transaction, and multiply by the interest rate for one period. The following formula can be used to calculate the interest for a single period:

balance * APR / 12 = interest

where APR stands for annual percentage rate and 12 represents the number of months in a year.

When we apply this formula to Samir's balance and APR, we get:

5336.22 * 0.29 / 12 = 128.95 in interest

As a result, the total new balance is:

5336.22 - 607 + 186 + 128.95 = 5044.17

We get the following when we round to the nearest hundredth:

$5,044.17

As a result, Samir now has a balance of $5,044.17.

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Answer: numbers 1,3 and 4

Step-by-step explanation:

The experimental probability that the card selected was a K or 6 is 17/100 or 0.17.

What is probability?

Probability is a measure of the likelihood or chance of an event occurring. It is expressed as a number between 0 and 1, where 0 indicates that an event is impossible and 1 indicates that it is certain. The probability of an event can be calculated by dividing the number of favorable outcomes by the total number of possible outcomes. In other words, it is the ratio of the number of desired outcomes to the total number of outcomes.

The frequency of card 6 is 7 and the frequency of card K is 12. However, the card K is also counted in the total count for JQK, so we need to subtract 2 from the frequency of K to get the actual count of K.

Actual count of K = 12 - 2 = 10

Total count of 6 and K = 7 + 10 = 17

The experimental probability of drawing a K or 6 is the frequency of drawing K or 6 divided by the total number of draws:

Experimental probability = (frequency of K or 6) / (total number of draws)

Experimental probability = 17 / 100

Therefore, the experimental probability that the card selected was a K or 6 is 17/100 or 0.17.

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Answer:

(y - 2)² = -12(x - 5)

Step-by-step explanation:

A parabola is a locus of points, which are equidistant from the focus and directrix;

Generic cartesian equation of a parabola:

y² = 4ax, where the:

Focus, S, is: (a, 0)

Directrix, d, is: x = -a

a > 0

Put simply, a is the horinzontal difference between the directrix and the vertex or between the vertex and focus;

Always a good idea to do a quick drawing of the graph;

We are the told the focus, F, is: (2, 2) and directrix, d, is: x = 8;

First thing to note, the vertex, or turning point will be in line with the focus vertically, i.e. they will share the same y-coordinate;

Horizonatally, it will be halfway between the focus and the directrix, i.e. halfway between 8 and 2;

Therefore, the vertex will be will be (5, 2);

We can also work out a:

a = 8 - 5 = 5 - 2

a = 3

Substituting this value of a into the generic cartesian equation:

y² = 4(3)x

y² = 12x

The focus and directrix will be:

S: (3, 0)

d: x = -3

Next thing to note, a parabola curves away from the directrix;

In this case, the directrix is x = 8, so the vertex will be the right-most point on the parabola, it will curve off to the left and the focus will also be to the left;

What we want to do is compare with y² = 12x;

This parabola, has a vertex (0, 0), which is the left-most point that curves off to the right and a focus also to the right;

Since we know the formula of this parabola, if we figure out how to transform it into the one in the question, we can find out it's equation;

What we should recognise first is that the parabola in the question is reflected in the y-axis, compared to y² = 12x;

So we apply the transformation that corresponds to this, i.e. use the f(-x) rule:

y² = 12(-x)

y² = -12x

Now the two graphs will have the same shape and orientation;

The focus and directrix will also be affected:

S: (-3, 0)

d: x = 3

Now, the only remaining difference would be the coordinates of the focus and directrix of the two graphs;

The focus of the graph in the question is 5 units to the right and 2 units upwards compared to the focus of y² = -12x;

The directrix is 5 units to the right of that of y² = -12x;

So we apply a translation transformation of 5 units right and 2 units up, like so:

(y - 2)² = -12(x - 5)

Replace y with (y - 2) to translate up 2 units;

Replace x with (x - 5) to translate 5 units right.

We know have a parabola with focus, (2, 2), directrix, x = 8 and vertex, (5, 2), i.e. the parabola in the question;

Hence, the equation of the parabola in the question is:

(y - 2)² = -12(x - 5)

It might seem a bit long and complicated to begin with, but can be done very quickly if you can get used to it.

The Davenports' medical expenses contribute to their total itemized deductions are $375 (7.5% for 2019).

The costs you incurred for state and local income or sales taxes, real estate taxes, personal property taxes, mortgage interest, and disaster losses are all included in itemised deductions. You can also count charitable donations and a portion of your out-of-pocket medical and dental costs.

Currently for the 2019 (due 2020), you can deduct medical expenses that exceed 7.5% of your AGI, but back then in 2019, the threshold was 7.5%, not 10%.

So the Davenports can only deduct

$5,250 - ($65,000 x 7.5%) = $375

if they decided to itemize their deductions.

The threshold will increase back to 10% starting 2020 (due 2021) tax returns.

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The values of x for which the series converges is x ∈ (-1/5, 1/5). The sum of the series for those values of x is (-5x)/(1 + 5x).

The series is [tex]\Sigma^{\infty}_{n=1}(-5)^nx^n[/tex].

We can write this series as [tex]\Sigma^{\infty}_{n=1}(-5x)^n[/tex].

This is a infinite geometric series with first term a = -5x and common ration r = -5x.

It is convergent when

|r| < 1

|-5x| < 1

|-5| |x| < 1

5|x| < 1

Divide by 5 on both side, we get

|x| < 1/5

The series is convergent when x ∈ (-1/5, 1/5).

Sum of the series is

Sₙ = a/1 - n

Sₙ = (-5x)/{1 - (-5x)}

Sₙ = (-5x)/(1 + 5x)

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The complete question is:

Find the values of x for which the series converges. Find the sum of the series for those values of x.

[tex]\Sigma^{\infty}_{n=1}(-5)^nx^n[/tex]

The joint probability density function for the random variables Y1 and Y2 E(Y1+Y2) and V(Y1+Y2) is 41/144.

To find E(Y1+Y2), we need to integrate the sum of Y1 and Y2 over their joint probability density function:

E(Y1+Y2) = ∫∫ (y1 + y2) f(y1,y2) dy1 dy2

= ∫∫ (y1 + y2) (2) dy1 dy2, where the limits of integration are 0 to 1 for both y1 and y2 and y1+y2 <=1

= ∫[tex]0^1[/tex] ∫[tex]0^{(1-y1)}[/tex](y1 + y2) (2) dy2 dy1

= ∫[tex]0^1[/tex] (2y1 + 1) (1-y1)² dy1

= 5/12

To find V(Y1+Y2), we can use the formula V(Y1+Y2) = E[(Y1+Y2)²] - [E(Y1+Y2)]².

First, we need to find E[(Y1+Y2)^2]:

E[(Y1+Y2)²] = ∫∫ (y1+y2)² f(y1,y2) dy1 dy2

= ∫∫ (y1² + y2² + 2y1y2) (2) dy1 dy2, where the limits of integration are 0 to 1 for both y1 and y2 and y1+y2 = 1

= ∫[tex]0^1[/tex] ∫[tex]0^{(1-y1)}[/tex] (y1² + y2² + 2y1y2) (2) dy2 dy1

= ∫[tex]0^1[/tex] (1/3)y1³ + (1/2)y1² + (1/2)y1

(1/3)y1 + (1/4) dy1

= 7/12

Next, we need to find [E(Y1+Y2)]²:

[E(Y1+Y2)]² = (5/12)² = 25/144

Therefore, V(Y1+Y2) = E[(Y1+Y2)²] - [E(Y1+Y2)]² = (7/12) - (25/144) = 41/144.

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Answer:

z=3.75

Step-by-step explanation: