For each of the following statements, determine whether it is true or false. If it is true, give a proof; if it is false, give a counterexample. (a) If X and Y are two random variables defined on the same sample space, 0≤ X(ω)≤Y(ω) for all ω∈Ω, and E(Y) exists, then E(X) exists. (b) If X and Y are two random variables defined on the same sample space, X(ω)≤ Y(ω) for all ω∈Ω, and E(Y) exists, then E(X) exists. (c) For any random variable X with S X
⊆[1,[infinity]), we have E( X
1
)= E(X)
1
. (d) If f and g are two pdfs, then fg is also a pdf. (e) If X is a continuous random variable with pdf f X
, then Y=∣X∣ is also a continuous random variable.
Answers
(a) False. A counterexample can be constructed as follows: Let X be a constant random variable such that X(ω) = 0 for all ω ∈ Ω, and let Y be a non-constant random variable such that Y(ω) = 1 for all ω ∈ Ω. In this case, 0 ≤ X(ω) ≤ Y(ω) holds for all ω ∈ Ω, but E(Y) exists while E(X) does not exist.
(b) True. If X(ω) ≤ Y(ω) for all ω ∈ Ω and E(Y) exists, then E(X) exists as well. To prove this, we can use the property of the expectation operator that states if X ≤ Y, then E(X) ≤ E(Y). Since X(ω) ≤ Y(ω) for all ω ∈ Ω, it follows that E(X) ≤ E(Y). Since E(Y) exists, it implies that E(X) exists as well.
(c) True. For any random variable X with S_X ⊆ [1, ∞), we can express X as the product of two random variables: X = X1, where X1 is the indicator random variable taking the value 1 if X ≥ 1 and 0 otherwise. Since E(X1) = E(X1^2) = 1 * P(X ≥ 1) = P(X ≥ 1), we have E(X1) = P(X ≥ 1). On the other hand, E(X) = E(X1) * E(X) = P(X ≥ 1) * E(X). Therefore, E(X1) = E(X)/P(X ≥ 1), which implies E(X1) = E(X)/P(X ≥ 1).
(d) False. The product of two probability density functions (pdfs) is not necessarily a pdf. To illustrate this, consider two pdfs f(x) = 2x for 0 ≤ x ≤ 1 and g(x) = 3x^2 for 0 ≤ x ≤ 1. The product fg(x) = 6x^3 does not satisfy the properties of a pdf since it does not integrate to 1 over its entire support [0, 1].
(e) False. The statement is not true in general. When taking the absolute value of a continuous random variable X, the resulting random variable Y = |X| may not be continuous. A counterexample can be constructed by considering X as a continuous random variable with a standard normal distribution (mean = 0, variance = 1). The absolute value of X, |X|, would yield a random variable that has a mixture distribution, taking the value 0 with positive probability and following the standard normal distribution elsewhere. Therefore, Y = |X| is not a continuous random variable in this case.
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Related Questions
What is the perimeter of triangleaeb? 16.4 ft 18.5 ft 18.7 ft 22.9 ft
Answers
Given the side lengths of the triangle, the perimeter of the triangle aeb is 18.5 ft
How to determine the perimeter?The side lengths of the triangle are given as:
5 ft, 5 ft and 8.5 ft
Add the side lengths to get the perimeter
Perimeter = 5 +5 +8,5
Evaluate the sum
Perimeter = 18.5
Hence, the perimeter of the triangle is 18.5
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If you are tossing a six-sided die, what is the probability of getting either a 3 or a 4 on your third toss and a 6 on your fourth toss?.
Answers
The probability to get a 3 or 4 in the third toss and 6 in the fourth toss is 1/18.
The sample space for a six-sided die is
{1, 2, 3, 4, 5, 6}
Hence the total number of possibilities = 6
The rolling of a die is IID that is Independently and Identically distributed.
Hence,
the result of one toss will not affect the result of the successive tosses.
Probability
= n(E)/n
= no of favorable outcomes for any event /total no. of outcomes
Here,
n = 6
Let A be the event of getting a 3 or 4 in the third toss
Let B be the event of getting 6 on the fourth toss
E(A) = {3,4}
n(A) = 2
Hence,
P(A) = 2/6
= 1/3
B = {6}
n(B) = 1
P(B) = 1/6
Since these are IID,
P(A and B) = P(A) X (B) [Property of independent events]
Hence P(A∩B) = 1/3 X 1/6
= 1/18
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Use the data to answer the following question.
Consider the mean, median, mode, and range of each data set. Which is NOT the same for both sets of data?
A.
mode
B.
range
C.
median
D.
mean
Answers
Answer: a
Step-by-step explanation:
Version A:
Mean = 20
Median = 20
Mode = 10 and 30
Range = 20
Version B:
Mean = 20
Median = 20
Mode = 20
Range = 20
What is median?It is the middles value of the given set of numbers after arranging the given set of numbers in order.
We have,
Version A:
Mean:
It is the average value.
= (5 x 10 + 3 x 15 + 20 + 3 x 25 + 5 x 30) / 17
= (50 + 45 + 20 + 75 + 150) / 17
= 20
Median:
It is the mid-value.
10, 10, 10, 10, 10, 15, 15, 15, 20, 25, 25, 25, 30, 30, 30, 30, 30
= 20
Mode:
The number that occurs most times.
= 10 and 30
Range:
Highest value - Lowest value.
= 30 - 10
= 20
Version B:
Mean:
= (10 + 3 x 15 + 5 x 20 + 3 x 25 + 30) / 13
= (10 + 45 + 100 + 75 + 30) / 13
= 20
Median:
The number that occurs most times.
10, 15, 15, 15, 20, 20, 20, 20, 20, 25, 25, 25, 30
= 20
Mode:
= 20
Range:
= 30 - 10
= 20
Thus,
Version A:
Mean = 20
Median = 20
Mode = 10 and 30
Range = 20
Version B:
Mean = 20
Median = 20
Mode = 20
Range = 20
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Tia is making strawberry lemonade for a baseball tournament. The recipe calls for 2
3
cup of strawberries for each gallon of lemonade. She has 6 cups of strawberries. How many gallons of lemonade can she make?
Tia can make
gallons of strawberry lemonade.
Answers
Answer:wow
Step-by-step explanation:
Answer:
4
Step-by-step explanation:
i am pretty sure i could be wrong
why is the term ""arc"" used to describe units of measure for angular distance
Answers
The term "arc" is used to describe units of measure for angular distance because it refers to the length of an arc on a circle.
Angular distance is the measure of the separation between two points on a circle or an arc. The term "arc" is used to describe this measure because it refers to the length of the arc connecting the two points on the circle.
This length is a fraction of the circumference of the circle and is proportional to the angle between the two points. The use of the term "arc" in this context is a nod to the geometric origins of angular measurement, which is based on the properties of circles and their angles.
The most common units of angular measurement are degrees, minutes, and seconds, which are all based on the division of a circle into 360 equal parts (degrees) and further subdivisions (minutes and seconds).
Other units of angular measurement, such as radians and gradians, are also based on the properties of circles and their angles.
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I don’t know how to answer this ?
Answers
Answer:
36 + 72 + 36 = ?
Step-by-step explanation:
Answer:
AB and AC are equal because the triangle is an isosceles triangle
Step-by-step explanation:
what is the measure of angle OAC
Answers
Answer:
60
Step-by-step explanation:
solve for x, Round to the nearest thenth. x,9.2,16.5
Answers
Steps:
16.5-9.2=7.3
9.2-7.3=1.9
A cup holder in a car contains 19 quarters, 39 dimes, some number of nickels, and 58 pennies. If all the coins in the cup holder equal $10.08, then how many nickels are in the cup holder?
Answers
Answer:
17 nickels
Step-by-step explanation:
To be able to find the answer, you can say that the sum of the value of each coin multiply for its quantity is equal to 10.08, which you can express as follows:
quarters= 0.25
dimes= 0.10
nickels= 0.05
pennies= 0.01
(0.25*19)+(0.10*39)+(0.05*x)+(0.01*58)=10.08, where
x= the quantity of nickels
Now, you can solve for x:
4.75+3.9+0.05x+0.58=10.08
0.05x=10.08-4.75-3.9-0.58
0.05x=0.85
x=0.85/0.05
x=17
According to this, the answer is that there are 17 nickels in the cup holder.
Please help! Correct answer only, please!
Fill in the missing numbers to complete the pattern:
___, ____, 5.1 , 5.2 , 5.3 , 5.4
Answers
Answer:
4.9,5.0
Step-by-step explanation:
Thier is two open spaces and the decimals are decreasing by 1 tenth. So if 1 tenth is being decreased the order would be 4.9, 5.0, 5.1, 5.2, 5.3, 5.4.
Answer:
4.9, 5 , 5.1 , 5.2 , 5.3 , 5.4
Step-by-step explanation:
5.4-5.3 = 0.1
5.3-5.2 = 0.1
5.2-5.1 = 0.1
then the pattern is : adding 0.1 to the number to get the next number
5.1 - 0.1 = 5
5 - 0.1 = 4.9
easy question but need answer quick!!
Answers
Answer:
Step-by-step explanation:
Step 1: Find slope of Linear function A
\(slope = \frac{y_2 - y_1}{x_2-x_1} = \frac{-5--1}{-4--2} = \frac{-5 +1}{-4+2} = \frac{-4}{-2} = 2\)
Step 2 : Find slope of B
Since they are perpendicular to each other, product of their slope = -1
That is,
\(m_A \times m_B = - 1\\2 \times m_B = -1\\\\slope, m_B = \frac{-1}{2}\)
Given the function passes through (4, -6)
Equation of the linear function :
\(x_3 = 4 \ , \ y_3 = -6\\\\(y - y_3) = m_B (x - x_3)\\\\(y - -6) = -\frac{1}{2} (x - 4)\\\\y + 6 = -\frac{1}{2}x + 2\\\\y = -\frac{1}{2}x + 2 - 6 \\\\y = -\frac{1}{2}x -4\)
Part B
Linear Function is parallel to B
Therefore the slope of B and slope of C are the same. And the equation will be of the form :
\(y = -\frac{1}{2} x + C\)
Observer X is standing on Plate C near the triple junction of Plates A,B and C. On the map, fill in the dashed plate margins with appropriate symbols, arrows and labels. In 25 years, where will the triple junction be located with respect to X ? where AVB=2 cm/yr (East) BVC=4 cm/yr (East) and note that the plates A and C are continental plates and the plate B is an oceanic plate.
Answers
Given that plate A is continental and plate B is oceanic, the eastward movement of the plate margins, plate A and plate B are both moving eastward while plate C remains stationary or moves at a slower rate.
To represent this on the map, we can use symbols, arrows, and labels as follows:
Plate A (continental plate): Draw an arrow pointing eastward with an appropriate symbol to represent a continental plate. Label it as "Plate A."
Plate B (oceanic plate): Draw an arrow pointing eastward with an appropriate symbol to represent an oceanic plate. Label it as "Plate B."
Plate C (continental plate): Draw a dashed line to represent the plate margin of plate C, indicating that it is stationary or moving at a slower rate compared to plates A and B.
After 25 years, considering the rates of plate motion given, the triple junction will have moved in the following direction:
Plate A will have moved 2 cm/yr (eastward) for 25 years, resulting in a total eastward movement of 50 cm.
Plate B will have moved 4 cm/yr (eastward) for 25 years, resulting in a total eastward movement of 100 cm.
Based on these movements, the triple junction will be located 50 cm east and 100 cm east of its current position with respect to observer X.
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Mary rates the books she reads on a scale of 0 to 10 stars, where 10 is the highest. the dot plot shows her ratings for the books she has read over the past two months. a dot plot plots book rating. 5 and 9 have 2 dots. 6 has 1 dot. 7 has 4 dots. 8 has 3 dots. 10 has five dots. after the data was recorded, mary gives her latest book a rating of 1 star. how will this new rating affect the mean rating of her books?
Answers
The new rating of 1 star lowers the mean rating of Mary's books from approximately 5.76 stars to 5.5 stars.
To determine how the new rating of 1 star will affect the mean rating of Mary's books, we need to calculate the mean rating before and after the addition of the new rating.
Before adding the new rating:
The mean rating is calculated by summing up all the ratings and dividing it by the total number of ratings. In this case, we can calculate the mean rating based on the given dot plot:
Number of books with a rating of 5: 2
Number of books with a rating of 9: 2
Number of books with a rating of 6: 1
Number of books with a rating of 7: 4
Number of books with a rating of 8: 3
Number of books with a rating of 10: 5
Total number of ratings = 2 + 2 + 1 + 4 + 3 + 5 = 17
Sum of ratings = (5 * 2) + (9 * 2) + (6 * 1) + (7 * 4) + (8 * 3) + (10 * 5) = 98
Mean rating = Sum of ratings / Total number of ratings = 98 / 17 ≈ 5.76
The mean rating before adding the new rating is approximately 5.76 stars.
After adding the new rating:
If Mary gives her latest book a rating of 1 star, we need to recalculate the mean rating by including this new rating.
Total number of ratings now becomes 17 + 1 = 18.
Sum of ratings now becomes 98 + 1 = 99.
Mean rating after adding the new rating = Sum of ratings / Total number of ratings = 99 / 18 ≈ 5.5
The new mean rating after adding the 1-star rating is approximately 5.5 stars.
Therefore, the new rating of 1 star lowers the mean rating of Mary's books from approximately 5.76 stars to 5.5 stars.
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Which of the following is the graph of y = StartRoot negative x minus 3 EndRoot?
Answers
Answer:
Use a graphing calc and plug in the equation.
Step-by-step explanation:
Answer:
It's C on EDGE
Step-by-step explanation:
A gardening box is 412 yd by 212 yd by 34 yd. What is the maximum amount of planting soil that can be used to fill the gardening box? 8716 yd³ 8316 yd³ 734 yd³ I don't know.
Answers
The amount of soil in the gardening box is an illstration of volume
The maximum amount of planting soil that can be used to fill the gardening box is 2969696 cubic yards
How to determine the amount of planting soil neededThe dimension of the gardening box is given as:
Length = 412 yd
Width = 212 yd
Height = 34 yd
Assume that the dimensions of the gardening box are not too thick.
The volume of the box is the product of its dimensions.
So, we have:
Volume = 412 yd * 212 yd * 34 yd
This gives
Volume = 2969696 cubic yards
Hence, the amount of soil that can be used to fill the gardening box is 2969696 cubic yards (none of the options is correct)
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Answer:
THE ANSWER IS DOWN BELOW :)
Step-by-step explanation:
The sum of 500 consecutive integers is 250.
what is the product of the same 500 numbers?
Answers
The required product of the same 500 consecutive integers whose sum is equal to 250 is zero.
Let us consider consecutive integers represented by x, x+ 1, x+ 2 , ..... so on.
Sum of 500 integers = 250
x + x + 1 + x+ 2 + ....+ x + 499 = 250
⇒ 500x + 1 + 2 + ...+499 = 250
⇒ 500x + ( 500 × 499 )/2 = 250
⇒ 500x + 250 × 499 = 250
⇒ 250 ( 2x + 499 ) = 250
⇒ 2x + 499 = 1
⇒x = -249
Required 500 numbers are :
-249 , -248 , -247 , ......, 0, 1, ....... 250
Here zero is also a part of 500 integers.
Product of 499 number with zero is zero.
⇒( -249 ×-248 ×.....× 250 ) × 0 = 0
Therefore, the product of 500 consecutive integers with sum 250 is equal to zero.
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can someone please help me please
Write the Inverse of the following statement. "If two angles have the same measure, then they are congruent."
Answers
Answer:
If two angles are congruent, then they have the same measure.
Step-by-step explanation:
Solve the system by graphing, then state the solution as an ordered pair
HELP ASAP!!!!!
Answers
Step-by-step explanation:
Simply graph the two equations....the intersection of the two graphs is the solution
The parking lot charges $2 for the first hour plus 50¢ for each additional half hour or part thereof. What is the total charge for parking a car in the lot from 11:30 a.m. until 2:15 p.m.?
Answers
Approximately, how many national parks are there in the united states? a. 25 b. 50 c. 120 d. 474 please select the best answer from the choices provided a b c d
Answers
Approximately, there are total 50 national parks is the United states.
The United States has approximately 50 national parks, which are congressionally designated procted areas operated by the National Park Service, an agency of the Department of the Interior.
National Parks are designated for their natural beauty, unique geological features, diverse ecosystems, and recreational opportitinies.
Some of the major national parks are Yesomite National Park, Bryce Canyon National Park, Acadia national park, National park of America Samona, Arches National park etc.
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Answer:
The answer is 50
Step-by-step explanation:
yes
Mai is flying a plane at an altitude of 1600 ft. She sights a stadium at an angle of
depression of 35°. What is mai’s approximate direct distance to the stadium?
Answers
Answer:
2,789ft
Step-by-step explanation:
The set up will give a right angled triangle.
The altitude will be the opposite side = 1600 ft
The angle of depression = 35°
approximate direct distance to the stadium is the hypotenuse of the trisngle= x
Using the SOH trig identity to get the distance
sin 35 = opp/hyp
sin 35 = 1600/x
x = 1600/sin35
x = 1600/0.5736
x = 2,789.4ft
Hence mai’s approximate direct distance to the stadium is 2,789ft
what would be the slope of (1,-2) and (3,0)
Answers
Answer:
1
Step-by-step explanation:
Hello :)
So to answer this problem, we can use the rise over run formula.
The rise is the difference of any two y values.
The run is the difference of the x values from the same two y values.
This question gives us 1,-2 and 3,0
We can use them for the formula:
-2 - 0 divided by 1 - 3
-2/-2 = 1
Answer:
\(m=1\)
General Formulas and Concepts:
Pre-Algebra
Order of Operations: BPEMDAS
Brackets Parenthesis Exponents Multiplication Division Addition Subtraction Left to RightAlgebra I
Slope Formula: \(m=\frac{y_2-y_1}{x_2-x_1}\)Step-by-step explanation:
Step 1: Define
Point (1, -2)
Point (3, 0)
Step 2: Find slope m
Substitute [Slope Formula]: \(m=\frac{0+2}{3-1}\)Evaluate Addition/Subtraction: \(m=\frac{2}{2}\)Evaluate Division: \(m=1\)What is the area of the figure?
Answers
Answer: 27 inches sq.
Step-by-step explanation:
3*3=9
2*9=18
18+9=27
(2x-45)
nearpod is very difficult for me, my apologies
Answers
Which of the following ranges is most appropriate to represent the numerical data on the vertical axis of a Bar Chart? OA) 0 to 60 OB) 10 to 20 OC) 0 to 55 OD) 30 to 60
Answers
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Answer:
C
Step-by-step explanation:
I took the quiz
Use Delta y = f'(x) Delta x to find a decimal approximation of the radical expression. 124 What is the value found using Delta y f' (x) Delta x? 124 = (Round to three decimal places as needed.)
Answers
Using Delta y = f'(x) Delta x to find a decimal approximation of the radical expression, we get the value found using Delta y f' (x) Delta x is 124.176.
The Delta y = f’(x) Delta x is the linear approximation formula used in calculus to estimate the value of an unknown function based on its derivative at a particular point.
In order to solve the problem using Delta y = f’(x) Delta x, we first have to rewrite the radical expression in terms of a function.
Let the function f(x) = sqrt(x) represent the radical expression.
Then, we will find the value of f'(x) by differentiating the function f(x) using the Power Rule of differentiation.
f(x) = sqrt(x) f'(x) = 1/2x^(-1/2)
Next, we will substitute the given value of x in the function f(x) and f’(x) to solve for Delta y = f'(x) Delta x.
Given, x = 124
Delta x = 2 Δy = f’(x) Δx = 1/2 * 124^(-1/2) * 2 = 0.1762
Hence, the decimal approximation of the radical expression is 124.176.
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Please help me. Please provide 5 examples of binomial equations
with solutions. thank you.
Answers
The answers are as follows:
1. 3x + 2 = 10, solution: x = 8/3, 2. 2y - 5 = -3, solution: y = 1, 3. 4a + 7 = 19, solution: a = 3, 4. 5b - 9 = 16, solution: b = 5, 5. 6c + 4 = -14, solution: c = -3.
Here are five examples of binomial equations along with their solutions:
1. Example: 3x + 2 = 10
Solution: Subtract 2 from both sides: 3x = 8. Divide both sides by 3: x = 8/3.
2. Example: 2y - 5 = -3
Solution: Add 5 to both sides: 2y = 2. Divide both sides by 2: y = 1.
3. Example: 4a + 7 = 19
Solution: Subtract 7 from both sides: 4a = 12. Divide both sides by 4: a = 3.
4. Example: 5b - 9 = 16
Solution: Add 9 to both sides: 5b = 25. Divide both sides by 5: b = 5.
5. Example: 6c + 4 = -14
Solution: Subtract 4 from both sides: 6c = -18. Divide both sides by 6: c = -3.
A binomial equation consists of two terms connected by an operator (+ or -) and an equal sign. To find the solution, we aim to isolate the variable term on one side of the equation. We do this by performing inverse operations.
Step 1: Start with the equation 3x + 2 = 10.
Step 2: Subtract 2 from both sides to isolate the term with the variable: 3x = 8.
Step 3: Divide both sides by 3 to solve for x: x = 8/3.
Repeat these steps for each example to obtain the solutions for the respective variables.
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In the drawing, what must be the coordinates of D to show ABCD is a square?
Answers
Rectangle TUVW is on a coordinate plane at T (a, b), U (a 2, b 2), V (a 5, b − 1), and W (a 3, b − 3). What is the slope of the line that is parallel to the line that contains side TU?.
Answers
The slope of the line that is parallel to the line that contains side TU is -1/2. Answer: The slope of the line that is parallel to the line that contains side TU is -1/2.
Rectangle TUVW is on a coordinate plane at T (a, b), U (a 2, b 2), V (a 5, b − 1), and W (a 3, b − 3). We need to find the slope of the line that is parallel to the line that contains side TU.
To find the slope of a line, we use the slope formula:
Slope formula: slope
(m) = (y₂ - y₁) / (x₂ - x₁)
We can use the coordinates of points T and U to find the slope of the line TU.
Slope of line TU: slope
(m) = (y₂ - y₁) / (x₂ - x₁)
slope (m) = (b2 - b) / (a2 - a)
Or we can use the coordinates of points V and W to find the slope of the line VW, which is parallel to TU, since they are opposite sides of a rectangle.
Slope of line VW: slope (m) = (y₂ - y₁) / (x₂ - x₁)slope
(m) = (b-3 - (b-1)) / (a3 - a5)slope
(m) = (b-3 + 1) / (a3 - a5)slope
(m)
= (b-2) / (-2).
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What is the square root of 148?
Answers
Answer: 12.16
Step-by-step explanation:
Answer: 12.1655250606
Step-by-step explanation: this is the square root of 148 but if you round to a whole number it is 12