Let Z= max (X, Y) and W = min (X, Y) are two new random variables as functions of old random variables X and Y. (a). Determine fz (z) and fw (w) in terms of marginal CDFs of X and Y random variables, by first drawing the region of interest on X and Y plane. (b). Let x and y be independent exponential random variables with common parameter A. Define W = min (X, Y). Find fw (w).
Answers
(a) fz (z) and fw (w) in terms of cumulative distribution functions (CDFs) are:
fz(z) = Fx(z) * (1 - Fy(z)) + Fy(z) * (1 - Fx(z))
fw(w) = 1 - fz(w)
(b) If X and Y are independent exponential random variables with parameter λ, then fw(w) = \(1 - e^{-2\lambda w}\) for w ≥ 0.
To determine fz(z) and fw(w) in terms of the marginal cumulative distribution functions (CDFs) of X and Y random variables, we need to consider the region of interest on the X-Y plane.
(a) Drawing the region of interest on the X-Y plane:
The region of interest can be visualized as the area where Z = max(X, Y) and W = min(X, Y) take specific values. This region is bounded by the line y = x (diagonal line) and the lines x = z (vertical line) and y = w (horizontal line).
Determining fz(z):
To find fz(z), we need to consider the cumulative probability that Z takes a value less than or equal to z. This can be expressed as:
fz(z) = P(Z ≤ z) = P(max(X, Y) ≤ z)
Since X and Y are independent random variables, the probability can be calculated using the joint CDF of X and Y:
fz(z) = P(max(X, Y) ≤ z) = P(X ≤ z, Y ≤ z)
Using the marginal CDFs of X and Y, denoted as FX(x) and FY(y), respectively, we can express fz(z) as:
fz(z) = P(X ≤ z, Y ≤ z) = P(X ≤ z) * P(Y ≤ z) = FX(z) * FY(z)
Determining fw(w):
To find fw(w), we need to consider the cumulative probability that W takes a value less than or equal to w. This can be expressed as:
fw(w) = P(W ≤ w) = P(min(X, Y) ≤ w)
Since X and Y are independent random variables, the probability can be calculated using the joint CDF of X and Y:
fw(w) = P(min(X, Y) ≤ w) = 1 - P(X > w, Y > w)
Using the marginal CDFs of X and Y, denoted as FX(x) and FY(y), respectively, we can express fw(w) as:
fw(w) = 1 - P(X > w, Y > w) = 1 - [1 - FX(w)][1 - FY(w)]
Special case when X and Y are independent exponential random variables with parameter A:
If X and Y are independent exponential random variables with a common parameter A, their marginal CDFs can be expressed as:
\(FX(x) = 1 - e^{-Ax}\\FY(y) = 1 - e^{-Ay}\)
Using these marginal CDFs, we can substitute them into the formulas for fz(z) and fw(w) to obtain the specific expressions for the random variables Z and W.
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Related Questions
The table of values below represents a linear function and shows the amount of money in Juanita’s savings account since she began her part-time job. What is her monthly rate of savings?
Amount in Juanita’s Savings Account
Number of Months Working at Part-Time Job
0
2
4
6
8
Amount in Savings Account (in dollars)
$36
$60
$84
$108
$132
a.$12 per month
b.$18 per month
c.$24 per month
d.$36 per month
Answers
Answer:
The answer is C., $24 per month.
Answer:
A $12
Step-by-step explanation:
she has saved $36 before she gets a part time job, after 2 months, her savings have increased from 36 to 60. 2 months = 60 - 36, 2 months = 24. now we divide 24 by 2, and the answer is 12. option A is correct.
14 and 1/8 divided by 10 please help
Answers
Answer:
should be 1 and 33/80
Step-by-step explanation:
Answer: 1 33/80
Step-by-step explanation:
the area of a right-angle triangle is 24m2.the base is 8 cm. find the length of it's hypotenuse in cm2
Answers
10
Explanation:
To find the area of a triangle you need to do 1/2 b x h. So we find out the height by halving 8 which is 4 and dividing it by 24. 24 divided by 4 is 6. Now we have found out the height which is 6 we need to find the hypotenuse by using Pythagoras (a2 + b2 = c2). 8 squared is 64 and 6 squared is 36. We add these values together to find c2. 36 + 64 = 100. Now we square root it to find the hypotenuse which is 10.
Is triangle LMN a right triangle?
Answers
It is demonstrated in the following figure itself that △LMN is a right triangle.
What is a right triangle?
A right triangle, also known as a right-angled triangle, right-perpendicular triangle, orthogonal triangle, or formerly rectangle triangle, is a triangle with one right angle, or two perpendicular sides.
The foundation of trigonometry is the relationship between the sides and various angles of the right triangle.
A right triangle's hypotenuse is its longest side, its "opposite" side is the one that faces a certain angle, and its "adjacent" side is the one that faces the angle in question.
To describe the sides of right triangles, we utilize specific terminology.
The side opposite the right angle is always the hypotenuse of a right triangle.
So, in the given figure, it is shown that △LMN is a right triangle.
Therefore, it is demonstrated in the following figure itself that △LMN is a right triangle.
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You have been saving $12 each week for many weeks. One day, you decide to count your savings and find that you have $384. Write and solve a multiplication equation to find how many weeks w you have been saving.
Answers
Answer:
Hope I helped! Brainleist!
Step-by-step explanation:
you get $12/week
and you have $
your equation is
y=12x
now...
y = 384
so
384=12x
x= 32
On a coordinate plane, a line goes through (0, 3) and (3, negative 1). A point is at (negative 3, 2).
What is the equation of the line that is parallel to the given line and passes through the point (−3, 2)?
3x − 4y = −17
3x − 4y = −20
4x + 3y = −2
4x + 3y = −6
A given line has the equation .
What is the equation, in slope-intercept form, of the line that is perpendicular to the given line and passes through the point (0, 9)?
Answers
Answer: D: 4x+3y=-6
Step-by-step explanation:
Answer:
4x + 3y = −6
Step-by-step explanation:
just took the test!
A 12- V battery powers a series circuit that contains two lightbulbs. The voltage across one of the lightbulbs is 8 V. What fractional part of the circuits total energy is transformed in the second lightbulb?
Answers
Answer:
1/3
Step-by-step explanation:
12-8=4
4/12 = 1/3
calculated fields must always contain at least one constant. T/F
Answers
The given statement "calculated fields must always contain at least one constant" is false because calculated fields do not necessarily have to contain a constant value.
They can involve variables, formulas, functions, or expressions that perform calculations based on the given data. The purpose of calculated fields is to derive new values or perform computations based on existing data within a system or application.
While constants can be used in calculated fields, they are not a requirement. The specific requirements and structure of calculated fields may vary depending on the context and the software or system being used.
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given the mean of gre scores is 150 and the standard deviation is 8.8. what proportion of combined gre scores can be expected to be between 155 and 160?
Answers
The proportion of combined gre scores can be expected to be between 155 and 160 is 0.1574
The mean of gre score = 150
The Standard deviation = 8.8
The proportion of combined gre score expected to be between 155 and 160 can be found using the normal distribution,
P(155 < x < 160)
Let us first find their standard score using the formula,
Z = (X -μ) / σ
where Z is the standard score
X is the random sample
μ is the mean
σ is the standard deviation
For x = 155,
Z = 155 - 150 / 8.8
= 5 / 8.8
= 0.57
For x = 160,
Z = 160 - 150 / 8.8
= 10/8.8
= 1.14
P(155 < x <160) = P( 0.54 < z < 1.14)
By using the z-table , we could find the value of P(0.54 < z < 1.14 ) will be 0.1574.
Therefore , the combined gre score between 155 and 160 is 0.1574.
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What is the slope of the line that passes through the points (3, 4) and (0,−2)?
Answers
Answer:
2
Step-by-step explanation:
divide the change in y by the change in x
4-(-2)/ 3-0 = 2
3. a spherical balloon is inflated so that the volume is increasing at the rate of 5 m^3/min. at what rate is the radius increasing when the volume is 36πm^3? v =4πr^3 / 3
Answers
Main Answer: The rate at which the radius is increasing is 1/3 m/min when the volume is 36πm³.
Supporting Explanation: Given that the volume of the spherical balloon is increasing at the rate of 5 m³/min and the volume of the balloon is given as 36π m³.The volume of the balloon is given by v = 4πr³ / 3. Therefore, 4πr³ / 3 = 36π. This gives the radius of the balloon as: r = cuberoot(27) = 3m.Then, differentiating the volume equation with respect to time, t, we get: dv/dt = 4π(3r²)dr/dt. Rearranging the above equation, we get: dr/dt = (1/3)(dv/dt) / r².Substituting the values, we get: dr/dt = (1/3 * 5) / 3²dr/dt = 1/3 m/min. Therefore, the rate at which the radius is increasing is 1/3 m/min when the volume is 36πm³.Here, the keywords used are spherical balloon, volume, rate, radius, and cuberoot.
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▶ Find the work done by the constant force F = 5¡ + 2j – 3k in moving an object in the space on the straight line connecting the points P(1,2,3) and Q = (1, −1, 4).
Answers
The work done by the constant force F = 5i + 2j - 3k in moving an object along the straight line connecting the points P(1, 2, 3) and Q(1, -1, 4) in space is zero.
To calculate the work done by a force along a straight line, we use the formula:
Work = Force · Displacement
Given the force vector F = 5i + 2j - 3k and the displacement vector from P to Q, which can be calculated as D = Q - P = (1 - 1)i + (-1 - 2)j + (4 - 3)k = 0i - 3j + 1k = -3j + k, we can calculate the dot product of F and D:
Work = (5i + 2j - 3k) · (-3j + k)
The dot product is calculated by multiplying the corresponding components of the two vectors and summing them:
Work = (5 * 0) + (2 * -3) + (-3 * 1) = 0 - 6 - 3 = -9
Since the work done is equal to -9, it means that the force F does negative work in moving the object along the straight line from P to Q. In other words, the force is acting in the opposite direction of the displacement, resulting in zero net work done.
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What does surface analysis chart show?
Answers
Answer:
Step-by-step explanation:
A surface analysis chart is a type of weather map that displays information about the current pressure and wind patterns over a particular region. The chart provides a snapshot of the weather conditions at a specific point in time and helps meteorologists to identify and track the location, movement, and development of high- and low-pressure systems, fronts, and other weather features.
The surface analysis chart typically shows a map of the Earth's surface with various meteorological symbols and information, including isobars (lines of equal pressure), fronts (boundaries between different air masses), and wind direction and speed. The chart may also include information about temperature, precipitation, and cloud cover.
Surface analysis charts are used in weather forecasting, air traffic control, and other applications where an understanding of current weather conditions is important. They provide important information for short-term weather forecasting, such as predicting the location of fronts and the development of thunderstorms, and for long-term weather planning, such as determining the most suitable routes for shipping or aviation.
Drag the term to the corresponding probability. Each term may be used only once. CertainLikelyUnlikelyImpossible Probability Term Probability < 12 Probability = 0 Probability = 1 Probability > 12
Answers
Thus, the probability are Zero represents the impossibility.
1/2 equals probable and unlikely
1 = event happens.
Because we don't know how something will turn out, we might talk about the probability of one outcome or the potential for several outcomes.
Typically, the likelihood is stated as the proportion of favourable events to all other potential outcomes in the sample space.
The probability of an event is calculated using the formula P(E) = (Number of favourable outcomes) (Sample space).
Given that probability can only be a number between 0 and 1,
The zeroth value is the impossible.
so there is a 50% chance for both likely and unlikely.
If 1, the event must take place.
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Question 2(Multiple Choice Worth 5 points)
(03.05 LC)
Which of the following is the equation of the ellipse with a vertical major axis, center at (1, -3), a = 7, and b = 5?
○ (x-1)²+(y+32²-1
25
49
0 (x+12+(y-32²-1
25
49
O(y-322²(x+12²
49
25
O(y+32 (x-12
49
25
+
1
-1
Answers
The equation of the the ellipse with a vertical major axis, center at (1, -3), a = 7, and b = 5 is \(\frac{(x-1)^2}{49}+\frac{(y+3)^2}{25}\)
Equation of the ellipses:
The Equation of the ellipse with center at (h, k)
\(\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}\)
where,
a = length of semi-major axis
b = length of semi-minor axis
Given,
Here we have the value of center as (1, -3) and the values of a = 7 and b = 5.
Here we need to find the equation of the ellipse.
According to the formula we we have the following values are given,
We know the value of center of ellipses is written as,
(h, k) = (1, -3)
And the values of
a = 7 and b = 5
Therefore, when we apply the values on the formula then we get,
\(\frac{(x-1)^2}{7^2}+\frac{(y-(-3))^2}{5^2}\)
When we simplify the term then we get,
\(\frac{(x-1)^2}{49}+\frac{(y+3)^2}{25}\)
Therefore, the equation of the ellipse is \(\frac{(x-1)^2}{49}+\frac{(y+3)^2}{25}\).
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Shelby made equal deposits at the beginning of every 3 months into an RRSP. At the end of 9 years, the fund had an accumulated value of $55,000. If the RRSP was earning 3.50\% compounded monthly, what was the size of the quarterly deposits? Round to the nearest cent
Answers
The size of the quarterly deposits in Shelby's RRSP account was approximately $147.40.
Let's denote the size of the quarterly deposits as \(D\). The total number of deposits made over 9 years is \(9 \times 4 = 36\) since there are 4 quarters in a year. The interest rate per period is \(r = \frac{3.50}{100 \times 12} = 0.0029167\) (3.50% annual rate compounded monthly).
Using the formula for the future value of an ordinary annuity, we can calculate the accumulated value of the RRSP fund:
\[55,000 = D \times \left(\frac{{(1 + r)^{36} - 1}}{r}\right)\]
Simplifying the equation and solving for \(D\), we find:
\[D = \frac{55,000 \times r}{(1 + r)^{36} - 1}\]
Substituting the values into the formula, we get:
\[D = \frac{55,000 \times 0.0029167}{(1 + 0.0029167)^{36} - 1} \approx 147.40\]
Therefore, the size of the quarterly deposits, rounded to the nearest cent, is approximately $147.40.
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if 3 + √5/ 2√5 + 3 = a + b√5, find the values of rational numbers a and b.
Answers
Answer: The given question is solved :
Step-by-step explanation:
correct solution in imaje
A project has five activities with the durations (days) listed
below:
Activity
Precedes
Expected
Duration
Variance
Start
A, B
-
-
A
C
40
0.31
B
E
32
0.25
C
D
21
0.35
Answers
The critical path is the path with the longest duration, which in this case is A -> B -> D -> E with a duration of 11 days.
To determine the critical path of the project, we need to find the longest path of activities that must be completed in order to finish the project on time. This is done by calculating the earliest start time (ES) and earliest finish time (EF) for each activity.
Starting with activity A, ES = 0 and EF = 4. Activity B can start immediately after A is complete, so ES = 4 and EF = 7. Activity C can start after A is complete, so ES = 4 and EF = 6. Activity D can start after B is complete, so ES = 7 and EF = 9. Finally, activity E can start after C and D are complete, so ES = 9 and EF = 11.
The variance for each activity is also given, which allows us to calculate the standard deviation and determine the probability of completing the project on time. The critical path is the path with the longest duration, which in this case is A -> B -> D -> E with a duration of 11 days.
Using the expected durations and variances, we can calculate the standard deviation of the critical path. This information can be used to determine the probability of completing the project on time.
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Please help!!
It due tonight
Answers
Answer:
D. y = -8 and y = -10
Step-by-step explanation:
The y-intercept can be found by finding the value of y when x = 0.
According to the table, y-intercept will be between 2 points (6,-8) and (-1,-10)
=> the answer will be D
Antonio works at a grocery store. He puts 23 pounds of flour into 5 sacks. He put the same weight of flour into each sack. How much flour did Antonio put into each sack?
Answers
What is the growth rate for the linear function y=mx+b?
Answers
The growth rate for the linear function y=mx+b will be m or slope. the correct option is A.
What is a linear equation?An equation is said to be linear if the maximum power of the variable is consistently 1. Another name for it is a one-degree equation.
The equation of a line is given as below:-
y = mx + c
Where m is the slope and c is the y-intercept.
Slope or the gradient is the number or the ratio which determines the direction or the steepness of the line. The slope of the line actually determines the rate of change of the function.
Therefore, the growth rate for the linear function y=mx+b will be m or slope. the correct option is A.
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How many even numbers less than 500 can be formed using 1,2 3 4 5?
Answers
28 different even numbers can be formed using 1,2,3,4 and 5 that are less than 500.
To form the even number from a combination of five different numbers 1 to 5 and the even number less that will be formed from such a combination are less than 500. Therefore, in this scenario, 28 different event numbers will be formed. The cases to formed the even number from 1,2,3,4 and 5 are given below:
Case 1: There is only one digit in the number.
The only even integers in this situation are 2 and 4, adding up to a total of 2.
Case 2: There are exactly two digits in the number.
In this instance, the first digit must be one of the other four permitted digits and the last digit must be either 2 or 4, for a sum of 2*4=8.
Case 3: There are exactly three digits in the number.
In this instance, the middle digit must be one of the other four permitted digits, and the last digit must either be 2 or 4.
The total will be 2*3*2=12 if the middle digit is not 5, which means the first digit must be one of two digits other than the final two digits and 5.
As a result, there exist 28 even numbers with the digits 1, 2, 3, and 5 that are less than 500.
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HELP!!!!!!!!!!!!!!!!!!!!!!
Answers
The derivative of the function 4 / x - 1 / y = 3 is equal to y' = 4 / (4 - 3 · x)².
How to find the derivative of a function by two methods
In this problem we must use two differentiation methods in a function to obtain its first derivative. These methods are (i) explicit differentiation and (ii) implicit differentiation. Each procedure is described below:
Explicit differentation
Clear y in terms of x.Use derivative rules.Simplify the resulting expression.Implicit differentiation
Use derivative rules. Clear y' in terms of x and y.Substitute y in the resulting expression.Simplify the resulting expression.Explicit differentiation
Step 1
4 / x - 1 / y = 3
1 / y = 4 / x - 3
1 / y = (4 - 3 · x) / x
y = x / (4 - 3 · x)
Step 2
y' = [(4 - 3 · x) - x · (- 3)] / (4 - 3 · x)²
Step 3
y' = 4 / (4 - 3 · x)²
Implicit differentiation
Step 1
- 4 / x² + [1 / (y²)] · y' = 0
Step 2
(1 / y²) · y' = 4 / x²
Step 3
y' = 4 · y² / x²
Step 4
y' = 4 · [x / (4 - 3 · x)]² / x²
y' = 4 / (4 - 3 · x)²
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Hello people ~
Factorise: x^2 + 7x + 12
Answers
Step-by-step explanation:
x^2 + (4+3)X + 12
x^2 + 4x + 3X + 12
X (X + 4) + 3 ( X + 4)
( X + 3) (X +4)
Solving steps:
\(\sf \rightarrow x^2\:+\:7x\:+\:12\)
breakdown the following
\(\sf \rightarrow x^2\:+\:3x + 4x\:+\:12\)
break the expressions into groups
\(\sf \rightarrow \left(x^2+3x\right)+\left(4x+12\right)\)
factor the following
\(\rightarrow \sf x\left(x+3\right)+4\left(x+3\right)\)
factor the common term: (x+3)
\(\rightarrow \sf \left(x+3\right)\left(x+4\right)\)
The number of people is inversely proportional to the number of days for a bag of rice to last. If it takes 8 people for a bag of rice to last 5 days, how many days would it take 10 people?
Answers
Answer:
4
Step-by-step explanation:
the ratio is 8/5, so the inverse would be 5/8, then it went up to 10, so it would be 10/4
Plzzzzz help plzzzzz help meeee
Answers
K
BD bisects ZABC. Solve for x and find mZABC.
m/ABD = (6x), m/DBC = (2x+12)°
X=
m/ABC=
Answers
bisects x = -12AB / (2AB - 6BD)
m∠ABC = 6x
= 6 × (-12AB / (2AB - 6BD))
To solve for x and find the measure of angle ABC (m∠ABC), we will apply the angle bisector theorem and use the given information.
According to the angle bisector theorem, the ratio of the lengths of the segments created by an angle bisector is equal to the ratio of the measures of the angles formed by the bisector.
Let's set up the equation using the given information:
m∠ABD = 6x (angle ABD)
m∠DBC = 2x + 12 (angle DBC)
Using the angle bisector theorem, we have:
AB/BD = m∠ABD/m∠DBC
Since BD bisects ∠ABC, we can substitute the given measures into the equation:
AB/BD = (6x) / (2x + 12)
To solve for x, we can cross-multiply:
AB × (2x + 12) = BD × (6x)
Expanding both sides of the equation:
2ABx + 12AB = 6BDx
Rearranging the equation:
(2AB - 6BD)x = -12AB
Now we can isolate x:
x = -12AB / (2AB - 6BD)
The measure of angle ABC (m∠ABC), we substitute the value of x back into the expression:
Simplifying this expression further would require additional information about the lengths of AB and BD.
Without this information, we cannot find the exact value of m∠ABC.
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A store sells four printers for every five computers. The store sells 40 computers on Saturday. How many printers did it sell on Saturday?
Answers
P:C
4:5
_:40
40/5=8
4x8=32
the sscp exam consists of ____ multiple-choice questions, and must be completed within three hours.
Answers
Answer:
125 questions
Answer:
125 questions
Step-by-step explanation:
Graph the function.
h(x) = -1/5x^2+2x
Answers
Answer:
Hello there I would love to assist but could you give me a little more info I think I know the answer but I want to make sure I got it 100% correct unless you cant give me more info ill just tell you what I think it is but if u can give me more info before I give u the answer to make sure its correct would be appreciated
Step-by-step explanation: