Answer:
14 bows
Step-by-step explanation:
.5 yards is need for one bow if you have 7 yards you have 14(.5) yards
easier way a half goes into 7, 14 times since 14 divide by two is 7
Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution.
-x+6y=9
-4x+24y=48
Answer:
no solution
Step-by-step explanation:
- x + 6y = 9 → (1)
- 4x + 24y = 48 → (2)
multiplying (1) by - 4 and adding to (2) will eliminate x and y
4x - 24y = - 36 → (3)
add (2) and (3) term by term
0 + 0 = 12
0 = 12 ← false statement
this false statement indicates the system has no solution
Which of the following is the correct order when multiplying 2 binomials
A. First, outer, inner, last
B. Inner, first, last, outer
C. Last, inner, outer, first
D. Outer, last, first, inner
Answer:
A. First, outer, inner, last.
Step-by-step explanation:
€100 was invested in a savings account. After 10 years, there is
€300 in the account.
Work out how much money would have been in the account after
only 5 years if it had been gathering
a) annual simple interest.
b) annual compound interest.
Give each of your answers to the nearest €1.
c) Will the savings account have more money in it after 15 years if
it is gathering annual simple interest or annual compound interest?
Answer:
Simple Interest
SI=PRT/100
P=€100
SI=€300-€100 = €200
T=10 years
rate = unknown
Rate = SIx100/(pxt)
Rate = 200x100/(100x10)
= 20%
b)Compound interest
CI = P(1+i)ⁿ
CI= €300
P =€100
n = 10
€300 = €100(1+i)¹⁰
take the tenth root of both sides
€1.77 = €1.58(1+i)
divide both sides by 1.58
1.12 = 1 + i
i = 1.12 - 1
i = 12%
c)si=prt/100
=(100x20x15)/100
= €300
total = 300+100
€400
CI = P(1+i)ⁿ
= 100(1+0.12)¹⁵
=€547.35
It is gathering more money with annual compound interest
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A circular arc has measure 8 ft and is intercepted by a central angle of 2.9 radians. Find the radius r of the circle.
Do not round any intermediate computations, and round your answer to the nearest tenth.
The radius of the circle is 2.8feet (nearest tenth)
What is length of an arc?Arc length is the distance between two points along a section of a curve.
The length of an arc is expressed as ;
L = (tetha) / 360 × 2πr
L is the length of the arc
tetha is the angle made by the two radii and the arc.
In radian 360° = 2π
therefore ;
8 = 2.9/ 2π × 2πr
8 = 2.9 r
r = 8/2.9
r = 2.76 ft
therefore the radius of the circle is 2.8 feet( nearest tenth) .
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a significance test about a proportion is conducted using a significance level of . the test statistic equals . the p-value is . a. if were true, for what probability of a type i error was the test designed? b. if this test resulted in a decision error, what type of error was it?
a) The test was designed to have a 5% chance of making a Type I error. b) Reject H0 since p-value < significance level. c) Type I error (false positive) if H0 is true and rejected.
A significance test is used to determine whether there is sufficient evidence to reject a null hypothesis, which is a statement about a population parameter, such as a proportion, mean, or standard deviation. The significance level, often denoted by α, is the probability of making a Type I error, which is rejecting the null hypothesis when it is actually true. The commonly used significance level is 0.05, which means that the test is designed to have a 5% chance of making a Type I error.
In this scenario, the sample proportion is 0.12 and the p-value is 0.03, which is the probability of observing a sample proportion as extreme as 0.12 or more extreme, assuming that the null hypothesis is true. Since the p-value is less than the significance level, we have strong evidence against the null hypothesis, and we reject it. Therefore, we conclude that the proportion is significantly different from the hypothesized value.
If the null hypothesis is actually true, and we reject it based on the sample data, we have made a Type I error. In other words, we have falsely concluded that there is a difference when there is not. False positive is another name for this mistake. Conversely, a Type II error occurs when we fail to reject the null hypothesis when it is actually false, and this error is also known as a false negative.
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The complete question is:
A significance level of 0.05 is used when conducting a proportion-related significance test. 0.12 is the sample statistic. P-value equals 0.03.
a) What chance of a Type I error was the test designed for, if H0 were true?
b) What judgement would you draw on this test (reject or fail to reject)?
c) What kind of error, if any, was there as a result of this test?
Tyler built a toy box in the shape of a rectangular prism with an open top. The diagram below shows the toy box and a net of the toy box. What is the surface area, in square inches, of the toy box?
Answer:482 in ^2
Step-by-step explanation:
help me solve this equation
The area of the shaded portion is 6x^2
How to determine this?
The area of the shaded portion = Area of big rectangle - Area of small rectangle.
To get the area of big rectangle = L * B
A = 2x * 4x
A = 8x^2
To get area of small rectangle = L * B
A = 2x * x
A = 2x^2
To get the area of shaded portion ,
= 8x^2 - 2x^2
= 6X^2
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X=a+b+4 and a directly proportional to y squared , b is inversely proportional to y when y=2 x=18 when y=1 x=-3 find x when y=4
X = 11 when y = 4 when X=a+b+4 and a directly proportional to y squared , b is inversely proportional to y.
Given that a is directly proportional to y^2 and b is inversely proportional to y, we can write the following two relationships:
[tex]a = k * y^2[/tex]
[tex]b = m / y[/tex]
where k and m are constants. From the first equation, we can find the value of k when y = 2:
[tex]a = k * 2^2[/tex]
[tex]k = a / 4[/tex]
And from the second equation, we can find the value of m when y = 2:
[tex]b = m / 2[/tex]
[tex]m = 2b[/tex]
Substituting these values into the equation X = a + b + 4, we can find X when y = 2:
[tex]X = (k * 2^2) + (m / 2) + 4[/tex]
[tex]X = (a / 4) * 4 + 2b + 4[/tex]
[tex]X = a + 2b + 4[/tex]
[tex]X = 18[/tex]
We can now use the value of k and m to find X when y = 4:
[tex]X = (k * 4^2) + (m / 4) + 4[/tex]
[tex]X = (a / 4) * 16 + (2b) / 4 + 4[/tex]
[tex]X = 4a + 2b + 4[/tex]
Substituting the value of X = 18 when y = 2, we can solve for a and b:
18 = 4a + 2b + 4
14 = 4a + 2b
7 = 2a + b
Solving the system of equations for a and b, we find:
a = 5
b = 2
Finally, substituting these values into the equation X = a + b + 4, we can find X when y = 4:
X = (5) + (2) + 4
X = 11
Therefore, X = 11 when y = 4.
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A city's population is currently 365,000. If the population doubles every 24 years, what will the population be 96 years from now?
Answer: 5,840,000
Step-by-step explanation:
96/24=4
365,000x2x2x2x2=5,840,000
or
365,000 x 2^4=5,840,000
SST Grocery Store has a display of 12-ounce cans of soft drink in cases of 24. Each case
measures 16.00 inches by 10.75 inches by 5.00 inches. How many cubic inches are needed if 360
cases are on display?
OA 619,200 cubic inches
OB 309,600 cubic inches
OC 61,920 cubic inches
OD 123,840 cubic inches
The answer is 309,600 cubic inches, which is option B.
What is the volume?
Volume is the total amount of space inside a three-dimensional shape. Volume is measured in cubic units, such as “cubic inches” or in³.
Volume is the amount of space matter takes up. The matter is anything that has mass and takes up space. Volume is used to describe the size of an object.
The volume of each case is 16.00 inches * 10.75 inches * 5.00 inches = 880 cubic inches.
So, the volume of 360 cases is 360 * 880 cubic inches = 309,600 cubic inches.
Therefore, the answer is 309,600 cubic inches, which is option B.
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A college basketball player makes 80% of his free throws . Over the season he will attempt 100 free throws assume free throw attempts are independent , and let X be the total number of free throws he makes . a) The mean of X is ? Which on is correct -Cannot be determined / 80 /100 /0.80 ? b) The standard deviation of X is ? Which on is correct= 16/ 4/ 80/ 20? c) The probability that the basketball player makes at least 90 of these attempts is approximately ? Which on is correct= 0.0062/ 0.9938/ 0.2660 ? d) If the basketball player instead attempts only 10 free throws, the probability that he makes at most 4 of these is ? Which on is correct= 0.0009/ 0.9991/ 0.0064/ 0.9936?
A collegiate basketball player hits 80% of his free throw attempts. Assuming free throw attempts are independent, he will try 100 free throws this season the answers are as follows:
a) The mean of X is 80.
b) The standard deviation of X is 4.
c) The probability that the basketball player makes at least 90 of these attempts is approximately is 0.0062.
d) If the basketball player instead attempts only 10 free throws, the probability that he makes at most 4 of these is 0.0064.
As per the question given,
a) The mean of X is given by the formula: mean = n * p, where n is the number of trials and p is the probability of success on each trial. In this case, the player attempts 100 free throws with a probability of success of 0.8, so the mean is:
mean = 100 * 0.8 = 80
Therefore, the correct answer is 80.
b) The standard deviation of X is given by the formula: standard deviation = sqrt(n * p * (1 - p)). Substituting the values, we get:
standard deviation = sqrt(100 * 0.8 * 0.2) = 4
Therefore, the correct answer is 4.
c) To find the probability that the player makes at least 90 free throws, we can use the normal approximation to the binomial distribution. The mean is 80 and the standard deviation is 4, so we can standardize the value of X as follows:
z = (90 - 80) / 4 = 2.5
Using a standard normal distribution table, we can find the probability that Z is greater than 2.5, which is approximately 0.0062.
Therefore, the correct answer is 0.0062.
d) If the player attempts only 10 free throws, the number of successful attempts X follows a binomial distribution with n=10 and p=0.8. The probability that he makes at most 4 of these attempts is:
P(X <= 4) = P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4)
Using the binomial distribution formula, we can calculate each of these probabilities and sum them up. Alternatively, we can use a binomial probability table or a calculator to find the cumulative probability.
Using a binomial probability calculator, we get:
P(X <= 4) = 0.0064 (rounded to four decimal places)
Therefore, the correct answer is 0.0064.
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measures of variability are used to group of answer choices describe the extent to which scores in a distribution differ from one another. describe the extent to which scores in a distribution differ from the mode. make appropriate decisions regarding graph size, shape, and style. describe measures of central tendency.
Yes, measures of variability are used to describe the extent to which scores in a distribution differ from one another.
Yes, measures of variability are used to describe the extent to which scores in a distribution differ from one another.
Measures of variability include the range, variance, and standard deviation, and they give us a sense of how spread out the scores in a distribution are. These measures help us understand the variability or dispersion of the data and how far the individual scores in a distribution deviate from the mean or other measures of central tendency.
Measures of central tendency, such as the mean, median, and mode, describe the central or typical value of a set of scores. They provide a summary of the data and a single value that represents the center of the distribution.
The decisions regarding graph size, shape, and style are usually based on the type of data being presented and the message being conveyed. A histogram, for example, may be used to visually display the distribution of a set of scores, and the size, shape, and style of the graph can be adjusted to highlight important features of the data.
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Which choice is the explicit formula for the following geometric sequence?
0.3,-0.06, 0.012, -0.0024, 0.00048, ...
A. an=0.3(0.2)"
B. an=-0.2(0.3)(n-1)
C. an= -0.3(-0.2)(n-1)
D. an= -0.3(0.2)(n-1)
E. an = 0.3(-0.2)(n-1)
The formula for given sequence is aₙ = 0.3(-0.2)ⁿ⁻¹ i.e. E.
What exactly is a sequence?
A sequence is a collection of numbers that follows pattern. Olivia, for example, has been offered a position with a beginning monthly pay of $1000 and a yearly raise of $500. Can you figure out her monthly pay for the first three years? The prices will be $1000, $1500, and $2000. Note that Olivia's income over a number of years creates a sequence as they follow a pattern where the numbers increase by $500 each time.
Now,
Given sequence is 0.3,-0.06, 0.012, -0.0024, 0.00048, ...
Of all given options only E satisfies the given sequence because
for E
a₁=0.3*-0.2¹⁻¹
=0.3*1
=0.3
a₂=0.3*-0.2²⁻¹
=-0.3*0.2
=-0.006
a₃=0.3*-0.2³⁻¹
=0.3*0.04
=0.012
hence,
The formula for given sequence is aₙ = 0.3(-0.2)ⁿ⁻¹.
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Question 6: What are the vertex and range of y = |x - 2 + 1? *
Answer:
Step-by-step explanation:
The vertex of the equation y = |x - 2 + 1| is (2, 1). The range of the equation depends on the value of x. When x < 2, the equation becomes y = -x + 1, which gives us a line with a slope of -1 that intersects the y-axis at (0,1) and has a minimum value of 1. When x >= 2, the equation becomes y = x - 1, which gives us a line with a slope of 1 that intersects the y-axis at (0,-1) and has no minimum value. Therefore, the range of the equation y = |x - 2 + 1| is [1, infinity).
Sara can read 19 pages an hour. How many pages can Sara read in 95 min? Round to the nearest whole number.
Answer:
Step-by-step explanation:
Convers 95 min into hours
95/60 = 1.58 Hours
Sara can read 19 pages per hour
Total pages in 95 min or 1.58 hours =
19 * 1.58 = 30.02
Rounding off to nearest whole number- 30
Hope it helps
what is the slope of the line that contains the points (-2,6) and (6,-3)?
PLEASE SHOW WORK
Answer: [tex]-\frac{9}{8}[/tex]
Step-by-step explanation:
If you are given two points of a line, the slope is defined as rise/run. Rise is the difference in y coordinates, (think about it, going UP is RISING), and Run is the difference in x coordinates, (again its pretty intuitive.)
So Slope = [tex]\frac{y_{2} - y_{1}}{x_{2}-x_{1}}[/tex]
This can be calculated with the coordinates of the two points, (order wont matter youll get the same answer).
Slope = (-3-6)/(6-(-2)) = -9/8
OR, (going the other way)
Slope = (6-(-3))/(-2-6) = 9/-8 = -9/8
Easy!
I need help with this
The angle relationship that will fill the correct angle are given below.
What is an angle relationship?An angle is said to be generated when two or more lines intersect at a point. Thus series of angles which have some common relationship on comparison are formed. Some common angle relationships are; supplementary angles, complementary angles, vertical angles, etc.
The angle relationship to fill the correct angle are:
1. <AXE and <CXD are vertical angles.
2. <AXF and <DXF are supplementary angles.
3. <DXC and <BXC are complementary angles.
4. <CXB and <AXB are adjacent angles.
5. <AXC and <CXD are supplementary angles.
6. <DXE and <AXC are vertical angles.
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What happens if the residual is negative?
When the residual is negative it represents the predicted value is very much high.
A residual represents the measure of how well a line fits for the given individual data points which were plotted on the graph. Residual plot is representation on the graph which shows the residuals on the vertical axis also independent variable on the horizontal axis of the graph.If the residual value is positive it represents the predicted value is very low on the regression line.If the residual value is negative it represents the predicted value is very high on the regression line.Therefore, the negative residual value on the regression line represents the predicted value is very much high.
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this homework due now pls help i've been out sick
Answer: A. [tex]92cm^{3}[/tex]
Step-by-step explanation:
[tex]V= \frac{4}{3} \pi r^{3}[/tex]
[tex]V=\frac{4}{3}\pi 2.8^{3}[/tex]
[tex]4/3=1.33333333333[/tex]
calculate in calculator
[tex]1.33333333333\pi 2.8^3=91.9523225755[/tex]
round to nearest ones (9 tells 1 to go to 2)
Therefore, A. 92cm^3
Somebody please look at this picture and help me pleaseeeeeee
Answer:
I think it's
Step-by-step explanation:
usually involves reading the pattern on the table while keeping in mind that x goes on top and y on
Chandra just spends 15 minutes away or math problems she spends the same amount of time on each problem how many minutes does she spend on each problem
Chandra spends 3 minutes on each math problem if she spends a total of 15 minutes on math problems and spends the same amount of time on each problem
Chandra spends the same amount of time on each math problem. Let's call the amount of time she spends on each problem "t".
We know that Chandra spends a total of 15 minutes on math problems. If she spends the same amount of time on each problem, then the total time spent on all the problems is equal to the number of problems multiplied by the time spent on each problem.
So, we can write the following equation:
total time spent on all problems = number of problems * time spent on each problem
We know that the total time spent on all problems is 15 minutes, so we can substitute this into the equation:
15 = number of problems * t
We don't know the number of problems, but we can solve for t by rearranging the equation:
t = 15 / number of problems
So, the amount of time Chandra spends on each math problem is equal to 15 divided by the number of problems.
For example, if Chandra works on 5 math problems, then the amount of time she spends on each problem is:
t = 15 / 5 = 3
So, Chandra spends 3 minutes on each math problem if she spends a total of 15 minutes on math problems and spends the same amount of time on each problem.
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three numbers a, b, and c between 0 and 1 are selected independently and at random. what is the probability that these numbers appear in increasing order?
The probability of selecting three numbers in increasing order is 1/3, which means that in one out of every three trials, we will select three numbers in increasing order.
Probability is a branch of mathematics that deals with the measurement of the likelihood or chance of an event occurring. In this problem, we are asked to find the probability of selecting three numbers in increasing order from a range of 0 to 1.
Let's compute the probability of selecting three numbers in increasing order. Since the total number of possible outcomes is infinite, we need to use a continuous probability distribution to compute the probability. In this case, we can use the uniform distribution, which assumes that each number between 0 and 1 is equally likely to be selected.
The probability of selecting "a" as the smallest number is a/1 = a.
The probability of selecting "a" as the largest number is (1-a)/1 = 1-a.
The probability of selecting "a" as the middle number is 2(1-a)a, which is the product of the probability of selecting "a" and the probability of selecting "b" or "c" from the remaining range.
Therefore, the probability of selecting three numbers in increasing order is the sum of these probabilities:
P(increasing order) = ∫₀¹ (a * (1 - a) + (1 - a) * a + 2a(1 - a))da
P(increasing order) = ∫₀¹ 2a(1 - a)da
P(increasing order) = 2∫₀¹ a - a²da
P(increasing order) = 2[(a²/2) - (a³/3)] from 0 to 1
P(increasing order) = 2[(1/2) - (1/3)] = 1/3
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3=2x+y
4x+6=10y
Find what is x and what is y
Answer:
x=1 , y=1
Step-by-step explanation:
2x+y=3 - - (1)
4x+6=10y
=> 4x-10y=-6 - - (2)
Multiply eq (1) by 2, and we get ;
4x+2y=6 - -(3)
Now subtract eq (3) by eg (2)
4x+2y=6
-4x+10y=+6
12y=12
y=1
Sub value of y in eq (1)
2x+1=3
2x=2
x=1
X is a normally distributed random variable with mean 25 and standard deviation 9.
What is the probability that X is less than 16?
The required probability that X is less than 16 is approximately 0.15866 or 15.86%.
What is the Z -a score?A Z-score is stated as the fractional model of data point to the mean using standard deviations.
Here,
We can use the standard normal distribution to find the probability that a normally distributed random variable, X, is less than a certain value. To use the standard normal distribution, we need to standardize X by subtracting the mean and dividing by the standard deviation:
Z = (X - μ) / σ
In this case, X has a mean of 25 and a standard deviation of 9,
Z = (16 - 25) / 9
z = -1
To find the probability that X is less than 16, we need to find the corresponding value of Z using the standard normal distribution table or a calculator.
P(Z < -1) ≈ 0.15866
Therefore, the probability that X is less than 16 is approximately 0.15866 or 15.86%.
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Find the area of an isosceles triangle whose one side is 10 cm greater than each of its equal sides and perimeter is 100 cm.
Answer:
the area of the isosceles triangle is approximately 104.49 cm^2.
Step-by-step explanation:
The perimeter of the triangle is 100 cm, so we can write an equation using the lengths of the sides:
x + x + (x + 10) = 100
3x + 10 = 100
3x = 90
x = 30
So the equal sides of the triangle have length 30 cm and the side that is 10 cm greater has length 40 cm.
To find the area of the triangle, we can use the formula for the area of an equilateral triangle:
Area = (sqrt(3) / 4) * (side length)^2
Area = (sqrt(3) / 4) * 30^2
Area = (sqrt(3) / 4) * 900
Area = (sqrt(3) / 4) * 30 * 30
Area = (sqrt(3) / 4) * 900
Area = (30 * sqrt(3)) / 2
Area = 45 sqrt(3)
Joshua invested $97,000 in an account paying an interest rate of 6% compounded daily. Assuming no deposits or withdrawals are made, how much money, to the nearest cent, would be in the account after 8 years?
The required after 8 years, the investment will be worth approximately $156,753.
What is compound interest?Compound interest is the interest on deposits computed on both the initial principal and the interest earned over time.
Amount = principal[1 + r/n]ⁿᵃ
Principal = P rate= r Time =a , n = compounding frequency
Here,
To calculate the future value of Joshua's investment after 8 years with an interest rate of 6% compounded daily, we can use the formula:
In this case, P = $97,000, r = 6% = 0.06, n = 365 (since the interest is compounded daily), and t = 8.
So, we have:
[tex]A = $97,000 \times (1 + 0.06/365)^{365 * 8}[/tex]
A = $195,239.00
Therefore, after 8 years, the investment will be worth approximately $156,753.
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you leave your house, travel one mile due south, then one mile due east, then one mile due north. you are now back at your house ! where do you live? there is more than one solution; find as many as possible.
There are infinitely many possible locations for your house that would allow you to travel one mile south, one mile east, and one mile north and end up back at your starting point.
As per the question given,
This is a classic puzzle known as the "One-Mile Puzzle". There are actually infinitely many possible locations for your house that satisfy these conditions.
One way to think about it is to consider the fact that the one-mile distances you traveled form the sides of a triangle. Since you traveled one mile due south, one mile due east, and one mile due north, the triangle you formed is an isosceles right triangle, with the right angle at the point where you ended up.
If you let (x, y) be the coordinates of your house, then the point where you ended up after traveling one mile south, one mile east, and one mile north is (x, y-1) + (1, y) + (x, y+1), which simplifies to (2x+1, 2y).
So, any point (x, y) that satisfies the equation 2x + 1 = 2y will work as a possible location for your house. For example:
If x = 0, then y = 1/2, so your house could be located at (0, 1/2).
If x = 1, then y = 3/2, so your house could be located at (1, 3/2).
If x = -1, then y = -1/2, so your house could be located at (-1, -1/2).
And so on...
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4 ×(1 × – 10–3) Evaluate the expression.
Suppose C(x)= x^2−13x+74 for 0 ≤ x ≤25 represents the marginal cost in hundreds of dollars to produce an additional x thousand pens. Find how many additional thousand pens can be produced with marginal cost of no more than $5,200. Write the largest interval containing all possible answers
Answer: 78.39 thousand, (78,390)
Step-by-step explanation:
Using the quadratic formula with this gives us two answers, 78.39 and -65.39. Since a negative answer doesnt make sense in this context, we will go with 78.39 thousand pens, (78,390 pens). The limit on x however doesn't seem to make sense as it says x </= 25 and x>/= 0. The only two answers are both above and below this limit.
Let X denote the amount of time a book on two-hour reserve is actually checked out, and suppose the cdf is the following. F(x) = 0 x < 0 x2 25 0 ≤ x < 5 1 5 ≤ x Use the cdf to obtain the following. (If necessary, round your answer to four decimal places.)
(a) Calculate P(X ≤ 3).
(b) Calculate P(2.5 ≤ X ≤ 3).
(c) Calculate P(X > 3.5).
(d) What is the median checkout duration ? [solve 0.5 = F()].
(e) Obtain the density function f(x). f(x) = F '(x) =
(f) Calculate E(X).
(g) Calculate V(X) and σx. V(X) = σx =
(h) If the borrower is charged an amount h(X) = X2 when checkout duration is X, compute the expected charge E[h(X)].
By Using the Cumulative distributed Function(CDF)m we get:
a) P (x <= 3 ) = 0.36
b) P ( 2.5 <= x <= 3 ) = 0.11
c) P (x > 3.5 ) = 1 - 0.49 = 0.51
d) x = 3.5355
e) f(x) = x / 12.5
f) E(X) = 3.3333
g) Var (X) = 13.8891 , s.d (X) = 3.7268
h) E[h(X)] = 2500
Given:
The cdf is as follows:
F(x) = 0 x < 0
F(x) = (x^2 / 25) 0 < x < 5
F(x) = 1 x > 5
Now,
a) Evaluate the CDF given with the limits 0 < x < 3.
So, P (x <= 3 ) = (x^2 / 25) | 0 to 3
P (x <= 3 ) = (3^2 / 25) - 0
P (x <= 3 ) = 0.36
b) Evaluate the CDF given with the limits 2.5 < x < 3.
So, P ( 2.5 <= x <= 3 ) = (x^2 / 25) | 2.5 to 3
P ( 2.5 <= x <= 3 ) = (3^2 / 25) - (2.5^2 / 25)
P ( 2.5 <= x <= 3 ) = 0.36 - 0.25 = 0.11
c) Evaluate the CDF given with the limits x > 3.5
So, P (x > 3.5 ) = 1 - P (x <= 3.5 )
P (x > 3.5 ) = 1 - (3.5^2 / 25) - 0
P (x > 3.5 ) = 1 - 0.49 = 0.51
d) The median checkout for the duration that is 50% of the probability:
So, P( x < a ) = 0.5
⇒ (x² / 25) = 0.5
⇒ x² = 12.5
⇒ x = 3.5355
e) The probability density function can be evaluated by taking the derivative of the cdf as follows:
pdf f(x) = d(F(x)) / dx = x / 12.5
f) The expected value of X can be evaluated by the following formula from limits - ∞ to +∞:
E(X) = integral ( x . f(x)).dx limits: - ∞ to +∞
E(X) = integral ( x² / 12.5)
E(X) = x³ / 37.5 limits: 0 to 5
E(X) = 5³ / 37.5 = 3.3333
g) The variance of X can be evaluated by the following formula from limits - ∞ to +∞:
Var(X) = integral ( x² . f(x)).dx - (E(X))^2 limits: - ∞ to +∞
Var(X) = integral ( x³ / 12.5).dx - (E(X))^2
Var(X) = x⁴/ 50 | - (3.3333)^2 limits: 0 to 5
Var(X) = 5⁴/ 50 - (3.3333)^2 = 13.8891
Standard Deviation (s.d) (X) = sqrt (Var(X)) = sqrt (13.8891) = 3.7268
h) Find the expected charge E[h(X)] , where h(X) is given by:
h(x) = (f(x))^2 = x^2 / 156.25
The expected value of h(X) can be evaluated by the following formula from limits - ∞ to +∞:
E(h(X))) = integral ( x . h(x) ).dx limits: - ∞ to +∞
E(h(X))) = integral ( x^3 / 156.25)
E(h(X))) = x^4 / 156.25 limits: 0 to 25
E(h(X))) = 25^4 / 156.25 = 2500
Learn more about cumulative distribution function:
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