Given the random variable X with specific expected values and the equation Y = a + bx + cX, we are asked to find the correlation coefficient p(X,Y).
The correlation coefficient between two random variables X and Y is given by the formula:
p(X,Y) = Cov(X,Y) / sqrt(Var(X) * Var(Y))
To calculate the correlation coefficient, we need to find the covariance (Cov(X,Y)) and the variances (Var(X) and Var(Y)).
Given the expected values, we can calculate the required values as follows:
Cov(X,Y) = E[XY] - E[X]E[Y]
Var(X) = E[[tex]X^2[/tex]] - [tex](E[X])^2[/tex]
Var(Y) = E[tex][(Y - E[Y])^2][/tex]
Using the provided expected values, we can substitute them into the formulas:
Cov(X,Y) = E[XY] - E[X]E[Y] = E[(a + bx + cX)X] - (0)(E[a + bx + cX]) = E[aX + b[tex]X^2[/tex] + c[tex]X^2[/tex]] = a(E[X]) + b(E[[tex]X^2[/tex]]) + c(E[[tex]X^3[/tex]])
Var(X) = E[[tex]X^2[/tex]] - [tex](E[X])^2[/tex] = 1 - [tex](0)^2[/tex] = 1
Var(Y) = E[(Y - [tex]E[Y])^2[/tex]] = E[(a + bx + cX - [tex](E[a + bx + cX]))^2[/tex]] = E[[tex](a + bx + cX)^2[/tex]]
Using the provided values for E[[tex]X^3[/tex]] and E[[tex]X^4[/tex]], we can simplify the expressions further and calculate the values.
Once we have the values of Cov(X,Y), Var(X), and Var(Y), we can substitute them into the correlation coefficient formula to find p(X,Y).
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In ΔFGH, f = 930 inches, g = 520 inches and ∠H=169°. Find ∠G, to the nearest degree.
Answer:
Thats so hard oh my gosh!
Step-by-step explanation:
Answer:4 degrees
Step-by-step explanation:
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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For the following systems, draw a direction field and plot some representative trajectories. Using your graph, give the type and stability of the origin as a critical point. You may need to look at the eigenvalues to be sure. 3 5 3 -2 4 2 2 -2 1 x, b. X'= X, X 5 3 1 -5 4 1 4 2 2 a. X' c. X'= -63) — —
Plot direction fields and trajectories. Analyze eigenvalues to determine stability and type of critical point.
For system (a):
The direction field and trajectories should be plotted based on the given matrix:
[3 5] [x]
[3 -2] * [y]
To determine the type and stability of the origin as a critical point, we can analyze the eigenvalues of the matrix. The eigenvalues are found by solving the characteristic equation:
det(A - λI) = 0,
where A is the given matrix and λ is the eigenvalue.
For system (b):
The direction field and trajectories should be plotted based on the given matrix:
[1 -5] [x]
[4 1] * [y]
To determine the type and stability of the origin as a critical point, we can again analyze the eigenvalues of the matrix.
For system (c):
The direction field and trajectories should be plotted based on the given matrix:
[-6 3] [x]
[ -4 -2] * [y]
To determine the type and stability of the origin as a critical point, we once again analyze the eigenvalues of the matrix.
Analyzing the eigenvalues will allow us to determine if the critical point is a stable node, unstable node, saddle point, or any other type of critical point.
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One penny is 1% of a dollar
$ 0.25 is 25% of a dollars . $1.25 is 12.5% dollars
Answer:
$1.25 is NOT 12.5% dollars. It is 125% of a dollar.
Step-by-step explanation:
We have 1 and 1/4 dollars. 1 and 1/4 as a percentage is 125%
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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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
Please Help Quick I am being timed
A. 2
B. 3
C. 4
D. 5
Answer:
c.Step-by-step explanation:
As I figured out, any number that is not 0, meaning any number with a value is a Significant Digit. For example, in 3003 there's only 2 significant digits. The easiest way to figure these things out is adding the digits.
I hope this helps :D
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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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!*
if there's a how and a guy picks her up who's getting abused
Flip a coin 100 times. Find the expected number of heads, with its uncertainty (the typical fluctuation).
Roll a die 100 times. Find the expected number of times getting a '6', with its uncertainty.
Roll a die 100 times. Find the expected number times not getting a '6', with its uncertainty.
For each of the three cases:
Express the result (best value ± uncertainty) with uncertainty rounded to one significant digit.
Base your calculations, first, on the Binomial distribution.
Repeat the calculation but now based on the Poisson distribution.
Discuss the appropriateness of the Poisson distribution for these three cases. That is...
Does it seem to give a good approximation? Why should this be so?
Even if a good approximation, the Poisson cannot be quite right. Why not?
Hints: The Poisson approximates the binomial for n large and p small but allows infinite successes.
The typical fluctuation is based on the square root of the variance, which is equal to 5.
The typical fluctuation is based on the square root of the variance, which is equal to 3.7.
The typical fluctuation is based on the square root of the variance, which is equal to 3.7.
The Poisson distribution does not account for the finite probability of zero events occurring, and it can not handle problems where the expected number of events is very large, as it assumes an infinite number of successes, which is not the case in real life.
1) Binomial distribution
Binomial distribution is the probability distribution of obtaining exactly r successes in n independent trials with two possible outcomes of a given event.
The best value for the expected number of heads when a coin is flipped 100 times is 50.0, with the uncertainty being 5.0.
The typical fluctuation is based on the square root of the variance, which is equal to 100 x 0.5 x (1-0.5) = 25, which gives the fluctuation to be 5 (the square root of 25).
The Poisson distribution is an excellent approximation for binomial distributions, especially when n is large and p is small.
2) Binomial distribution
The best value for the expected number of times getting a '6' when a die is rolled 100 times is 16.7, with an uncertainty of 4.1.
The typical fluctuation is based on the square root of the variance, which is equal to 100 x (1/6) x (5/6) = 13.9, which gives the fluctuation to be 3.7 (the square root of 13.9)..
3) Binomial distribution
The expected number of times not getting a '6' when a die is rolled 100 times is 83.3, with an uncertainty of 4.1.
The typical fluctuation is based on the square root of the variance, which is equal to 100 x (5/6) x (1/6) = 13.9, which gives the fluctuation to be 3.7 (the square root of 13.9).
Poisson distribution
The Poisson distribution is a good approximation for the binomial distribution because n is large and p is small. Furthermore, the Poisson distribution can be used to predict the probability of a specific number of events occurring in a specific amount of time, which makes it ideal for modeling the number of radioactive decays or the number of phone calls to a call center.
However, the Poisson distribution does not account for the finite probability of zero events occurring, and it can not handle problems where the expected number of events is very large, as it assumes an infinite number of successes, which is not the case in real life.
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HELP ASAP I WILL GIVE BRAINLIEST IF I CAN
Answer: add 3
Step-by-step explanation:
Which form of a linear equation is defined by y = mx +b?
O A. Parallel form
O B. Standard form
O C. Slope-intercept form
O D. Point-slope form
The linear equation is defined by y = mx +b is called ''Slope-intercept form''.
What is Equation of line?The equation of line in point-slope form passing through the points
(x₁ , y₁) and (x₂, y₂) with slope m is defined as;
⇒ y - y₁ = m (x - x₁)
Where, m = (y₂ - y₁) / (x₂ - x₁)
Given that;
The linear equation is defined by,
⇒ y = mx + b
Now, We know that;
The equation defined as;
⇒ y = mx + b
Where, 'm' is slope and 'b' is y - intercept.
Hence, The linear equation is defined by y = mx +b is called ''Slope-intercept form''.
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Use the SIN to find x?
190m
Xm
1. Set Up
2. Operation
3. Answer
52.2
Answer:
sine=opposite/hypotenuse
sin52.2=x/150
sin0.790=x/150
x=118.5
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
Solve the system of equations below.
Leo has a rectangular garden with a perimeter of 40 feet. The width of the garden is 8 feet. What is the length of the garden?
Answer:
5 feet
Step-by-step explanation:
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.
Solve the equation on the interval [0,2m). √2 cos x + 1=0
the solution to the equation on the interval [0, 2π) is x = 3π/4.
To solve the equation, we want to find the values of x that satisfy the equation within the given interval.
First, we isolate the cosine term by subtracting 1 from both sides:
√2 cos(x) = -1
Next, we divide both sides by √2:
cos(x) = -1/√2
To find the solutions, we need to determine the angles whose cosine value is equal to -1/√2. We can use the unit circle or reference angles to determine these angles.
In the interval [0, 2π), the angle that satisfies cos(x) = -1/√2 is x = 3π/4.
Therefore, the solution to the equation on the interval [0, 2π) is x = 3π/4.
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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 the system of equations using substitution
y=x+1
x+y = 7
Write -8 7/8
as a decimal number,
Answer:
-8.875
Step-by-step explanation:
8 7/8 > -71/8 > -8.875
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] !!✎
▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁
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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To find the x-intercept, we let y = 0 and solve for x and to find y-intercept, we let x=0 and solve for y. Figure out the x-intercept and y-intercept in given equation of the line.
6x + 2y = 12
not bots with links or so help me i will
Answer:
X-intercept: (2, 0) Y-intercept: (0, 6)Step-by-step explanation:
X-intercept: y=0:
6x +2×0 = 12
6x = 12
x = 2
Y-intercept: x=0:
6×0 + 2y = 12
2y = 12
y = 6
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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A manufacturer claims that the lifetime of a certain type of battery has a population mean of μ = 40 hours with a standard deviation of a = 5 hours. Assume the manufactures claim is true and let a represent the mean lifetime of the batteries in a simple random sample of size n = 100. Find the mean of the sampling distribution of , μ = Find the standard deviation of the sampling distribution of , I What is P(40.6)? Round to the nearest thousandths (3 decimal places) The area this probability represents is (choose: right/left/two) tailed. Suppose another random sample of 100 batteries gives = 39.1 hours. Is this unusually short? (yes/no) Because P(≤39.1) = Round to the nearest thousandths (3 decimal places) The area this probability represents is
It should be noted that the probability of obtaining a sample mean of 39.1 hours or less is quite low (0.035), it can be considered unusually short.
How to calculate the probabilityThe mean of the sampling distribution of the sample mean, μ, is equal to the population mean, which is μ = 40 hours.
The standard deviation of the sampling distribution is σ(μ) = 5 / √100
= 5 / 10
= 0.5 hours.
Plugging in the values, we get (40.6 - 40) / 0.5
= 0.6 / 0.5
= 1.2.
Looking up the z-score of 1.2 in the standard normal distribution table (or using a calculator), we find that the probability is approximately 0.884.
Now, let's calculate P(≤39.1). Similarly, we calculate the z-score as (x - μ) / σ(μ), where x = 39.1 hours. Plugging in the values, we get (39.1 - 40) / 0.5
= -0.9 / 0.5
= -1.8.
Using the z-score table or a calculator, we find that the probability is approximately 0.035.
This probability represents the area under the curve to the left of -1.8, which is a left-tailed probability.
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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.
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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