The subset [28, 12, 11] is the only subset that adds up to 42 using backtracking.
To find a subset of the given numbers that adds up to 42 using backtracking, we can create a tree where each level represents a decision point of including or excluding a number from the subset.
Starting from the root node, we'll explore all possible combinations until we reach the desired sum or exhaust all possibilities.
Let's begin by constructing the tree:
[]
/ / \ \ \ \
29 28 12 11 7 3
/ \ / \ / \ |
28 12 11 12 11 11 7 7
/ \ | | |
12 11 12 11 7
/
11
Here, each node represents a number, and the path from the root to a particular node represents the chosen numbers for the subset. The root node ([]) represents an empty subset. We'll traverse this tree using backtracking to find the subset that adds up to 42.
Starting from the root node, we explore the left subtree by including the first number, 29:
[29]
/ / \ \ \ \
28 12 11 7 3
/ \ / \ / \ |
28 12 11 12 11 11 7 7
/ \ | | |
12 11 12 11 7
/
11
Since 29 is greater than the desired sum of 42, we backtrack to the root node and explore the right subtree by excluding 29:
[]
/ / \ \ \ \
28 12 11 7 3
/ \ / \ / \ |
28 12 11 12 11 11 7 7
/ \ | | |
12 11 12 11 7
/
11
Next, we explore the left subtree by including 28:
[28]
/ / \ \ \ \
12 11 7 3
/ \ / \ / \ |
12 11 7 11 7 7 3
/ \ | | |
11 7 7 7 3
/
11
Since 28 is less than the desired sum, we can explore its left subtree by including 12:
[28, 12]
/ / \ \ \ \
11 7 3
/ \ / \ / \ |
11 7 7 7 3
/ \ | | |
11 7 7 7 3
/
11
Since 28 + 12 = 40, which is still less than 42, we explore the left subtree of 12 by including 11:
[28, 12, 11]
/ / \ \ \ \
7 7 3
| | |
7 7 3
/
11
Now, we have reached a sum of 42 with the subset [28, 12, 11]. We have found a valid solution.
If we continue exploring the tree, we find that there are no other subsets that add up to 42.
Therefore, the subset [28, 12, 11] is the only subset that adds up to 42 using backtracking.
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Find the sample standard deviation. 12, 12, 12, 15, 18,
18, 18
show work pls
Answer:
Sample standard deviation = 3
Step-by-step explanation:
We can use the following steps to calculate the sample standard deviation for the given data set:
Step 1: First, calculate the mean of the data set by adding all the values and dividing by the total number of values:
(12 + 12 + 12 + 15 + 18 + 18 + 18) / 7 = 15.
Thus, the mean is 15.
Step 2: Next, subtract the mean from each value in the data set to get the deviation of each value from the mean.
(12 - 15) = -3
We can simply write -3 thrice since 12 appears thrice in the data set.(15 - 15) = 0
(18 - 15) = 3
Similarly, we can write 3 thrice since 18 appears thrice in the data set.Thus, the deviations are -3, -3, -3 0, 3, 3, and 3.
Step 3: Square each deviation:
(-3)^2 = 9
We can write 9 thrice since -3 appears as a deviation thrice.0^2 = 0
(3)^2 = 9
We can again write 9 thrice since 3 appears as a deviation thrice.Thus, the squares of all our deviations are 9, 9, 9, 0, 9, 9, and 9.
Step 4: Add up all the squared deviations:
9 + 9 + 9 + 0 + 9 + 9 + 9 = 54.
Thus, the sum of the squared deviations is 54.
Step 5: Find the variance by dividing the sum of squared deviations by one less than the number of values in the data set:
54 / (7 - 1) = 9
54 / 6 = 9
Thus, the variance is 9
Step 6: Finally, find the sample standard deviation by taking the square root of the variance:
√9 = 3
So, the sample standard deviation for this data set is 3.
Given that a is in Quadrant 2 and cos(a) = give an exact answer for the following: a sin(20) b. cos(2a) c. tan(20) = 2. Given that B is in Quadrant 4 and sin(B) = give an exact answer for the following: a sin(25) = b.cos(2B) c. tan(28) . Decimal approximations are not allowed for this problem, • Enter your answer in exact form. • Use "sqrt()" to represent.
a) In Quadrant 2, a sin(20) is equal to -sin(20).
b) In Quadrant 2, cos(2a) is equal to -cos(2a).
c) In Quadrant 2, tan(20) is equal to -tan(20).
In Quadrant 2, the angle 'a' is between 90 degrees and 180 degrees, or π/2 and π radians. Knowing that cos(a) is a negative value, we can determine the exact values for the given trigonometric expressions.
a) a sin(20) = -sin(20):
The sine function (sin) is positive in Quadrant 1 and negative in Quadrant 2. Therefore, the value of a sin(20) in Quadrant 2 is equal to the negative of sin(20). This means that the answer for a sin(20) is -sin(20).
b) cos(2a) = -cos(2a):
The cosine function (cos) is negative in Quadrant 2. Since we are given that a is in Quadrant 2, the angle 2a will also be in Quadrant 2. Therefore, cos(2a) in Quadrant 2 is equal to the negative of cos(2a). Thus, the answer for cos(2a) is -cos(2a).
c) tan(20) = -tan(20):
The tangent function (tan) is negative in Quadrant 2. Hence, the tangent of any angle in Quadrant 2 will be equal to the negative of its positive counterpart. Consequently, the answer for tan(20) in Quadrant 2 is -tan(20).
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During basketball practice, you make 8 free throws out of 20 shots taken. Using
experimental probability, how many free throws would you score out of 50
shots taken?
Answer:
if you make 4 free throws for every 10 shots taken you would make 20 out of 50 free throws
Step-by-step explanation:
HELP ASAP I WILL GIVE BRAINLIEST IF I CAN
Answer: add 3
Step-by-step explanation:
Nash uses his credit card with a 17.8% APR, compounded monthly, to pay for a cruise totaling $1,789.17. He can pay $650 per month on the card. What will the total cost of this purchase be?
Answer: 17.8% + $1.789.17 + $650 = 2439.348 or 2439
Step-by-step explanation: if you read the word problem it has a key word the key word is total.
Nash uses his credit card with a 17.8% APR, compounded monthly, to pay for a cruise totaling $1,789.17. He can pay $650 per month on the card. What will the total cost of this purchase be?
Alice is using a rope to pull a wagon. He exerts a force of 2 newtons at an angle of 48° from the floor.
Part A: Suppose the direction of the force changes from a 48° angle with the floor to a 70° angle with the floor. Determine the effect on the horizontal and vertical components of the force. Give forces to the nearest hundredth of a unit.
Part B: What implications does this have for pulling the wagon? Explain.
Answer:
Part A - i. 0.68 N ii. 1.88 N
Part B - The wagon will move slowly while being raised up off the ground.
Step-by-step explanation:
Part A: Suppose the direction of the force changes from a 48° angle with the floor to a 70° angle with the floor. Determine the effect on the horizontal and vertical components of the force. Give forces to the nearest hundredth of a unit.
Since the force of 2 N is exerted at this new angle of 70°,
i. the horizontal component u = 2cos70° = 0.684 N ≅ 0.68 N and its
ii. vertical component, v = 2sin70° = 1.879 N ≅ 1.88 N
Part B: What implications does this have for pulling the wagon? Explain.
Since we have a horizontal component of 0.68 N and a vertical component of 1.88 N, most of the force is used in raising the wagon off of the ground with less of it moving it forward. So, the wagon will move slowly while being raised up off the ground.
determine the normal and shear stresses at point d that act perpendicular and parallel, respectively, to the grains. the grains at this point make an angle of 35 ∘ with the horizontal as shown.
To determine the normal and shear stresses at point D, which act perpendicular and parallel to the grains, respectively, we require additional information such as the applied forces or loadings at that point.
The given question mentions the need to determine the normal and shear stresses at point D, which act perpendicular and parallel to the grains, respectively. However, to accurately calculate these stresses, we need more information about the system under consideration. Key details include the applied forces or loadings on the system, the material properties, and the specific orientation and arrangement of the grains at point D.
Normal stress, also known as axial stress, refers to the force per unit area acting perpendicular to the surface. Shear stress, on the other hand, represents the force per unit area acting parallel to the surface. The magnitudes and directions of these stresses depend on the applied forces, the geometry of the system, and the material properties.
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Can someone help me find this please and thank you:)
Answer:
The perimeter to the question is 30.
Step-by-step explanation:
.
[tex]3 + 3 + 3 + 3 + 9 + 9 \\ 6 + 6 + 18 \\ 12 + 18 \\ 30[/tex]
compute the accumulated value of $8600at 6.45% after 8
months (simple interest)
The accumulated value of $8600 at 6.45% after 8 months (simple interest) is $8971.90.
To compute the accumulated value of $8600at 6.45% after 8 months (simple interest), we need to use the formula for simple interest, which is given by:
I = P × r × t
Where, I is the interest earned, P is the principal amount, r is the interest rate, and t is the time in years.
Here, we have t in months, so we need to convert it into years by dividing by 12.
So, t = 8/12 = 2/3 years.
Now, substituting the given values, we get:
I = 8600 × 6.45/100 × 2/3 = $371.90
Therefore, the accumulated value of $8600 at 6.45% after 8 months (simple interest) is $8600 + $371.90 = $8971.90.
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Loretta is rolling an unfair 6 sided die with a single number between 1 and 6 on each face. She has a 70% chance of rolling a four. She is playing a game with a friend and knows that if she rolls a four on three of her next five rolls she will lose the game. She wants to determine the probability that she rolls a four on three of her next five rolls.
Which simulation design has an appropriate device and a correct trial?
Complete question is;
Loretta is rolling an unfair 6 sided die with a single number between 1 and 6 on each face. She has a 70% chance of rolling a four. She is playing a game with a friend and knows that if she rolls a four on three of her next five rolls she will lose the game. She wants to determine the probability that she rolls a four on three of her next five rolls.
Which simulation design has an appropriate device and a correct trial?
A) Using a fair coin let heads represent rolling a four and tails represent not rolling a four. Flip the coin five times.
B) Using a table of random digits select a digit between 0 and 9. Let 0-6 represent rolling a four and 7-9 represent not rolling a four. Select five digits.
C) Roll a fair die with a single digit between 1 and 6 on each face. Let four represent rolling a four and 1-3 and 5 and 6 represent not rolling a four. Roll the die five times.
D) Using a table of random digits select a digit between 1 and 6, ignoring digits 0, 7, 8, and 9. Let 4 represent rolling a four and 1-3 and 5 and 6 represent not rolling a four Select five digits.
Answer:
B) Using a table of random digits, select a digit between 0 and 9. Let 0-6 represent rolling a four and 7-9 represent not rolling a four. Select five digits.
Step-by-step explanation:
Since she knows that if she rolls a four on three of her next five rolls she will lose the game, then the best simulation that she will roll a four on three of the next five rolls will be option B because it uses a table of random digits and doesn't ignore any number but is well ordered with 0-6 representing a four and 7-9 not rolling a four.
if there's a how and a guy picks her up who's getting abused
Does the following argument illustrate the Law of Detachment?
Given: If the fuse has blown, then the light will not go on.
The fuse has blown.
Conclude: The light will not go on.
A. Yes
B. No
The correct answer is A. Yes. the argument conforms to the Law of Detachment.
Yes, the argument does illustrate the Law of Detachment. The Law of Detachment is a valid form of reasoning in propositional logic that states that if a conditional statement (p → q) is true and the antecedent (p) is true, then the consequent (q) can be inferred as true.
In the given argument:
The conditional statement "If the fuse has blown, then the light will not go on" can be represented as p → q, where p represents "the fuse has blown" and q represents "the light will not go on."
The given information states that the fuse has blown, which means that p is true.
According to the Law of Detachment, if p → q is true and p is true, we can conclude that q is also true. Therefore, we can infer that "the light will not go on" (q) is true.
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Use convolution notation with * and set up the integral to write the final answer of the following initial value ODE. There is no need to evaluate the integral. x" - 8x' + 12x = f(t)
The equation in integral form by taking the inverse Laplace transform: x(t) = L^-1((s^2 - 9s + 20)X(s)) + L^-1(F(s) + sx(0) + x'(0) - 8x(0)), where L^-1 represents the inverse Laplace transform.
To express the given initial value ordinary differential equation (ODE) using convolution notation with *, we can rewrite it as:
(x'' - 8x' + 12x) = f(t)
Taking the Laplace transform of both sides, we have:
s^2X(s) - sx(0) - x'(0) - 8(sX(s) - x(0)) + 12X(s) = F(s)
where X(s) is the Laplace transform of x(t), x'(0) is the initial condition of the derivative of x(t), x(0) is the initial condition of x(t), and F(s) is the Laplace transform of f(t).
Simplifying the equation, we get:
(s^2 - 8s + 12)X(s) - sx(0) - x'(0) + 8x(0) = F(s)
Now, let's express the left-hand side of the equation using convolution notation. Using the property that the Laplace transform of the derivative of a function f(t) is sF(s) - f(0), we can rewrite the equation as:
((s^2 - 8s + 12) - s + 8)X(s) = F(s) + sx(0) + x'(0) - 8x(0)
Simplifying further, we have:
(s^2 - 9s + 20)X(s) = F(s) + sx(0) + x'(0) - 8x(0)
Finally, we can express the equation in integral form by taking the inverse Laplace transform:
x(t) = L^-1((s^2 - 9s + 20)X(s)) + L^-1(F(s) + sx(0) + x'(0) - 8x(0))
where L^-1 represents the inverse Laplace transform.
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Which of the following equations could be the line of best fit?
A. x = 100
B. y = 100
C. y = x + 100
D. y = x - 100
The real answer is B
The scatter plot shows the amount of time Adam spent studying and his test scores, Predict Adam's test score when he studies for 6 hours
100
90
80
Test Score
70
O.
60
0
0 1 2 3 4 5
Time Studying (hours)
When he studies for 6 hours, Adam should score about
Answer:
I think 100
Step-by-step explanation:
3. if u get this right i’ll give u brainliest
Answer:
(-3,2)
Step-by-step explanation:
3/4 of eight is six so go to two and then find the x, as x wasn't on the numbers it is not a multiple of two, so three was the closest number
may be wrong
I need help please with this assignment
Answer:
a) 10 yards and 10[tex]\sqrt{3}[/tex] yards
Step-by-step explanation:
The intersection of the support beams in the middle form four right angles, since all four sides are equal. Because you know two angles of the triangle on the right hand side (90° and 30°), you can calculate the third angle (given that a triangle has 180°):
180°-90°-30° = 60°
This is a 30-60-90 triangle (if you don't know what that is, you can look it up). In a 30-60-90 triangle, the hypotenuse is 2x. In this case, the hypotenuse is 10 yards, so you can set up an equation given that information:
2x = 10
x = 5
Now that you know x of the 30-60-90 triangle, you can solve for the other two sides of the triangle.
The side directly across from the 30° angle is just x, which is equal to 5. The side directly across from the 60° angle is x[tex]\sqrt{3}[/tex], which in this case would be 5[tex]\sqrt{3}[/tex].
Because you want the full length of both beams, you'd multiple both sides by two to get the length of both entire beams:
5*2 = 10
(5[tex]\sqrt{3}[/tex])*2 = 10[tex]\sqrt{3}[/tex]
*I hope this makes sense!*
what is the quadratic equation for this graph? will make brainiest if also tell me how to put into vertex from
Answer:
Step-by-step explanation: The graph of a quadratic function is a parabola whose axis of symmetry is parallel to the y -axis. The coefficients a,b, and c in the equation y=ax2+bx+c y = a x 2 + b x + c control various facets of what the parabola looks like when graphed.
Hope you like it.
Solve for x
5=2x-3
Enter your answer in the box
x = __
Step-by-step explanation:
✧ [tex] \underline{ \underline{ \large{ \tt{G \: I \: V \: E \: N \: \:E \: Q \: U \: A \: T \: I \: O \: N}}}} : [/tex]
5 = 2x - 3❀ [tex] \underline{ \underline{ \large{ \tt{ \: T \: O \: \: F\: I\: N \: D}}} }: [/tex]
Value of x☄ [tex] \underline{ \underline{ \large{ \tt{S \: O \: L\: U \: T \: I\: O \: N}}}} : [/tex]
♨ [tex] \large{ \sf{5 = 2x - 3}}[/tex]
~Swap the sides of the equation :
⇾ [tex] \large{ \sf{2x - 3 = 5}}[/tex]
~We want to remove the 3 first. Since the original equation is -3 , we are going to use the opposite operation and add 3 to the both sides :
⇾ [tex] \large{ \sf{2x - 3 + 3 = 5 + 3}}[/tex]
⇾ [tex] \large{ \sf{2x \: \cancel{ - 3} \: \cancel{ + 3}}} = 8[/tex]
⇾ [tex] \large{ \sf{2x = 8}}[/tex]
~Now , We need to think about how to remove the coefficient 2. Since the opposite of multiplication is division , we are going to divide both sides of the equation by 2 :
⇾ [tex] \large{ \sf{ \frac{2x}{2} = \frac{8}{2}}} [/tex]
⇾ [tex] \boxed{ \large{ \sf{x = 4}}}[/tex]
☯ [tex] \underline{ \underline{ \large{ \tt{C \: H \:E \: C \: K}}}}: [/tex]
☪ [tex] \large{ \tt{L \: H \: S : \: 5 }}[/tex]
[tex] \large{ \tt{R \: H \: S \ \: : \: 2x - 3 = 2 \times 4 - 3 = 8 - 3 = \underline{ \tt{5}}}}[/tex] [ Plug the value of x ]
☥ Since this is a true statement , our answer ( x = 4 ) is correct. Yay! We got our answer :)
♕ [tex] \large{ \boxed{ \underline{ \large{ \tt{Our \: Final \: Answer : \boxed{ \underline{ \bold{ \text{x = 5}}}}}}}}}[/tex]
[tex] \underline{ \underline{ \text{H \: O\: P \: E \: \: I \: \: H \: E \: L \: P \: E \: D}}}[/tex] !! ♡
[tex] \underline{ \underline{ \text{H \: A \: V \: E \: A \: W \: O \: N \: D \: E \: R \: F \: U \: L \: D \: A\: Y \: / \: N\: I \: G\: H \: T}}}[/tex] !! ツ
☃ [tex] \underline{ \underline{ \tt{C \:A \: R \: R \: Y \: \: O \:N \: \: L \: E \: A \: R\: N \: I \: N \: G}}} [/tex] !!✎
▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁
Solve the following differential equation:
dV/dθ= V cotθ + V^3cosecθ
Comparing this with the left-hand side expression -V cosec^2θ - 3V^2cosec^2θcotθ, we can see that both sides are equal.
To prove the given equation, we will differentiate both sides with respect to θ and simplify the expression.
Differentiating V cotθ + V^3cosecθ with respect to θ:
d/dθ(V cotθ + V^3cosecθ) = d/dθ(V cotθ) + d/dθ(V^3cosecθ)
Applying the differentiation rules:
= V (-cosec^2θ) + 3V^2cosecθ(-cosecθcotθ)
= -V cosec^2θ - 3V^2cosec^2θcotθ
Now, we need to express the right-hand side of the original equation in terms of dV/dθ:
V cotθ + V^3cosecθ = V(cotθ + V^2cosecθ)
= V(cotθ + cotθV^2)
= Vcotθ(1 + V^2)
Taking the derivative of the right-hand side with respect to θ:
d/dθ(Vcotθ(1 + V^2)) = V(−cosec^2θ)(1 + V^2) + Vcotθ(0)
= -Vcosec^2θ(1 + V^2)
Comparing this with the left-hand side expression -V cosec^2θ - 3V^2cosec^2θcotθ, we can see that both sides are equal. Therefore, we have proved that dV/dθ = V cotθ + V^3cosecθ.
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Solve the proportion
17/4 = y/6
Answer:
This is your answer ☺️☺️
benjamin is making bow ties how many half yard long bow ties can he make if he has 18 feet of fabrics
To determine how many bow ties he can make, we need to convert the units of measurement and calculate the number of half-yard lengths in 18 feet.
To convert feet to yards, we need to divide the number of feet by 3, as there are 3 feet in a yard.
In this case, 18 feet is equal to 6 yards (18 feet ÷ 3).
Since Benjamin wants to make bow ties that are each half a yard long, we can calculate the number of bow ties he can make by dividing the total length of fabric (6 yards) by the length of each bow tie (0.5 yards).
Dividing 6 yards by 0.5 yards gives us a total of 12. Benjamin can make 12 half-yard long bow ties using the 18 feet of fabric he has.
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Tea costs £1.40 and coffee
costs 65p how much will 5
coffees and 2 teas cost?
Answer:
£6.05
Step-by-step explanation:
Multiply the number of coffees by the cost of coffee:
5 × 65p
325p
Multiply the number of teas by the cost of tea:
2 × £1.40
£2.80
Add the two answers together:
325p + £2.80
Convert p to £
£3.25 + £2.80
£6.05
find the slope of the passing line through the points -3/8 and 4/8
Answer:
Subtract the y's: 7 - 2 = 5
Subtract the x's in the same order: 6 - 4 = 2
slope = (difference in y)/(difference in x) = 5/2
Answer: 5/2
Use the formula to find the standard error of the distribution of differences in sample means, ¯x1−¯x2. Samples of size 120 from Population 1 with mean 81 and standard deviation 11 and samples of size 70 from Population 2 with mean 73 and standard deviation 17.
The standard error of the distribution of differences in sample means is approximately 2.8.
The standard error of the distribution of differences in sample means, ¯x1−¯x2, can be calculated using the formula:
SE(¯x1 - ¯x2) = sqrt[s1²/n1 + s2²/n2]
where, s1 and s2 are the standard deviations of the two populations, ¯x1 and ¯x2 are the sample means of the two populations, and n1 and n2 are the sample sizes of the two populations.
In this case,
Population 1 has a sample size of n1 = 120, a mean of ¯x1 = 81, and a standard deviation of s1 = 11.
Population 2 has a sample size of n2 = 70, a mean of ¯x2 = 73, and a standard deviation of s2 = 17.
Substituting these values into the formula,
SE(¯x1 - ¯x2) = sqrt[11²/120 + 17²/70]≈ 2.8
Therefore, the standard error of the distribution of differences in sample means is approximately 2.8.
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A 30-ft cable is stretched from the top of an antenna to an anchor point 12-ft from the base of the
antenna. How tall is the antenna? Round to the nearest tenth of a foot.
what is the slope and y intercept of -5x+3y=-9
Answer:
slope = 5/3
y-intercept = (0,-3)
Step-by-step explanation:
Make the equation in slope-intercept form:
-5x+3y=-9
add 5x on both sides
3y=5x-9
divide by 3 on both sides
y=5/3x-3
slope = 5/3
y-intercept = (0,-3)
find the value of the variable for each polygon
The value of variable h is,
⇒ h = 18
We have to given that,
A heptagon with angles are shown in figure.
Here, All the angles are,
⇒ 6h°, 132°, 146°, 146° , (6h + 10)° , (6h + 10)° , 146°, and 132°
We know that,
Sum of all the angles in a heptagon is, 900°
Hence, We get;
⇒ 6h° + 132° + 146° + 146° + (6h + 10)° + (6h + 10)° + 146° + 132° = 900°
Combine like terms,
⇒ 18h + 576 = 900
⇒ 18h = 900 - 576
⇒ 18h = 324
⇒ h = 18
Thus, The value of variable h is,
⇒ h = 18
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hi please help i’ll give brainliest
Answer:
Choice D
Step-by-step explanation:
I believe that it is Choice D. Due to the fact that the moons orbit around Earth is inclined around 5 degrees, not allowing an eclipse to happen every New or Full Moon. Most of the time, the Sun, Earth, and Moon don't line up precisely enough to cause an eclipse.
The joint density of X and Y is given by
f (x,y) = c1/2x^2y^2, 1
Compute c.
Therefore, the value of c is 2/9.
The joint density of X and Y is given by f (x,y) = c(1/2) x² y², 1.
We are to calculate the value of c.
Step-by-step solution: It is given that joint density of X and Y is f (x,y) = c(1/2) x² y², 1.
The joint probability density function f(x, y) satisfies the following properties:f(x, y) ≥ 0 for all x and y.f(x, y) is continuous in x and y.∫∞−∞∫∞−∞f(x, y)dxdy = 1
From the given joint density function, we can compute marginal density of X and marginal density of Y by integrating over the other variable, as follows: P(X = x) = ∫ f(x, y) dy and P(Y = y) = ∫ f(x, y) dx
Let's calculate the marginal density of X.P(X = x) = ∫ f(x, y) dy∫ f(x, y) dy = c(1/2) x² ∫y² dy Limits of integration are from -1 to 1.P(X = x) = c(1/2) x² [(1/3) y³]1 and -1.∫ f(x, y) dy = c(1/2) x² [(1/3) (1³ - (-1)³)]P(X = x) = c(1/2) x² [(1/3) (1 - (-1))]P(X = x) = c(1/2) x² (2/3)P(X = x) = (1/3) c x²P(X = x) = 1,
integrating over all possible values of x, we obtain:1 = ∫ P(X = x) dx= ∫ (1/3) c x² dx Limits of integration are from -1 to 1.1 = (1/3) c [(1/3) x³]1 and -1.∫ P(X = x) dx = (1/3) c [(1/3) (1³ - (-1)³)]1 = (1/3) c [(1/3) (1 - (-1))]1 = (1/3) c (2)1/2 = (2/9) c2/9 = c
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Answer:
Step-by-step explanation:
The joint density of X and Y is given by the function f (x,y) = c1/2x²y², 1. The value of c is 18.
We are required to find the value of c.
First, we need to know the definition of joint density.
A joint probability density function (PDF) is a statistical measure that describes the probability of two or more random variables occurring simultaneously in terms of their PDFs.
It's a measure of the probability of an event happening as a function of two variables, usually expressed as f(x,y).
So, in this problem, we have given the joint density of x and y is f(x,y) = c/2x²y², 1.
We can solve it by integrating it over the entire range.
The probability density function of X and Y can be found by integrating over the whole sample space.
[tex]$$\begin{aligned}&\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f_{X,Y}(x,y)dxdy=1\\&\implies\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f_{X,Y}(x,y)dxdy\\&=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}c\frac{1}{2}x^2y^2dxdy=1\\&=\frac{c}{2}\int_{-\infty}^{\infty}x^2dx\int_{-\infty}^{\infty}y^2dy\\&=\frac{c}{2}\cdot \frac{1}{3}\cdot \frac{1}{3}=1\end{aligned}$$[/tex]
Hence, [tex]$$\implies c=\boxed{18}$$[/tex].
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