K = 1/4, fY(y) = (8 - Y)/9
Random variables X and Y have a joint probability density function f(x,y) = K(2x+y) where 0<=x<=1, 0<=y<=2 and f(x,y) = 0 elsewhere. Also, Y = 2^(-X) + 3. Let's determine the value of K.
Determination of K
The probability density function f(x,y) must satisfy the following condition:
i.e., the integral of f(x,y) over the entire range of (x,y) should be equal to 1.
f(x,y) = 0 elsewhere implies that f(x,y) = 0 for x<0 and x>1 and y<0 and y>2. Hence, the range of integration should be [0,1] for x and [0,2] for y.
The integral of f(x,y) over the entire range of (x,y) can be expressed as follows:
[tex]∫∫K(2x+y)dydx = 1[/tex]
On integrating with respect to y first, we get:
[tex]∫(2x+y)dy = [2xy + (1/2)y^2][/tex]evaluated from 0 to 2
= 4x + 2
On integrating with respect to x, we get:
[2x^2 + 2x] evaluated from 0 to 1
= 4
On equating the integral value with 1, we get:
[tex]4K = 1K = 1/4[/tex]
Determination of probability density function of Y
We have [tex]Y = 2^(-X) + 3[/tex]. Therefore, for a given value of Y, the range of X can be determined as follows:
[tex]2^(-X) = Y - 3= > X = -log2(Y-3)[/tex]
Hence, the probability density function of Y can be obtained as follows:
[tex]fY(y) = ∫f(x,y)dxfY(y) = ∫f(x,2^(-X) + 3)dx[/tex]
From the given expression, we can observe that f(x,y) = 0 elsewhere implies that f(x,2^(-X) + 3) = 0 for x<0 and x>1 and y<3 and y>2. Also, the range of integration for x can be determined as follows:
For y<=3, X>=-log2(y-3). For y=2, the minimum value of X can be obtained by taking the limit as y tends to 2 from the right. The minimum value of X is therefore equal to [tex]-∞[/tex]. Therefore, the range of integration for x is [tex][-∞,1].[/tex]
fY(y) =[tex]∫f(x,2^(-X) + 3)dx = ∫(1/4)(2x + 2^(-X) + 3)dx[/tex]
fY(y) = (1/4)(x^2 - 2^(-X)x + 3x) evaluated from [tex]x=-∞ to x=1[/tex]
fY(y) = [tex](1/4)(1 - 2^(log2(Y-3)) + 3)[/tex]= (8 - Y)/9
Let's draw the probability density function of Y. The probability density function of Y is as follows:
fY(y) = (8 - Y)/9
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with 09) Let x, y be random variables joint probability density function f(x,y) = K (2x+ +y) of as 4 of y=2 443 Find K and (Hind : draw P CY<SX ) a picture )
Please help, Im stuck on this part of a review and Im really confused asap
Answer:
( 6, -1 )
Step-by-step explanation:
When you rotate 1 from the x axis by 90° it becomes -1 from the y axis.
When you rotate 6 by 9° from thr y axis, it becomes again 6 on the x axis
Your new x value is 6 and y is -1
So (6,-1)
Answer:
(-6, 1)
Step-by-step explanation:
To find the point obtained by rotating point P = (1, 6) counterclockwise by an angle of 90 degrees (r₉₀°), we can use the rotation formula:
x' = x * cos(θ) - y * sin(θ)
y' = x * sin(θ) + y * cos(θ)
In this case, θ is 90 degrees.
Substituting the values into the formula:
x' = 1 * cos(90°) - 6 * sin(90°)
y' = 1 * sin(90°) + 6 * cos(90°)
cos(90°) = 0 and sin(90°) = 1, so we have:
x' = 1 * 0 - 6 * 1 = -6
y' = 1 * 1 + 6 * 0 = 1
Therefore, r₉₀°(P) = (-6, 1). The point P = (1, 6) rotates counterclockwise by 90 degrees to the point (-6, 1).
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6(x-2)=-18 what does x=
Answer:
x = -1
Step by step explanation:
Distribute 6
6x-12= -18
Add 12 to both sides
6x= -6
Divide both sides by 6
x= -1
Hope this helps
Hang was trying to factor 10x^2 + 5x she found that the greatest common factor of these terms was 5x and made an area model what is the width of the area model
Width of Heng's area model is 2x + 1
Step-by-step explanation:
Given:
Greatest common factor is 5x
To Find:
The width = ?
Solution:
let the be the area
And 5x be the length
Then area = length x width
Now rewriting the formula for width, we get
Width =
Substituting the values in the above formula
Width =
Width = 2x + 1
Work out
1/8
of 760
please help
Answer: 95
Step-by-step explanation:
Think of 1/8 times 760 as 760/8 because it’s the same thing.
let x equals negative 14 times pi over 3 period part a: determine the reference angle of x. (4 points) part b: find the exact values of sin x, tan x, and sec x in simplest form. (6 points)
The exact values of sin x, tan x and sec x in simplest form are: sin x = -√3/2, tan x = √3, sec x = -2/√3.
To solve this problem, we'll work with the angle x, which is equal to -14π/3. Part a: Determine the reference angle of x. The reference angle is the positive acute angle formed between the terminal side of the angle and the x-axis. To find the reference angle, we ignore the negative sign and convert the angle to its equivalent positive angle within one revolution.
The positive equivalent angle can be obtained by adding 2π (or 360 degrees) repeatedly until we obtain a positive angle. In this case, we have: -14π/3 + 2π = -14π/3 + 6π/3 = -8π/3. The reference angle of x is 8π/3. Part b: Find the exact values of sin x, tan x, and sec x in simplest form. sin x = sin(-14π/3) = -sin(14π/3) (Using the symmetry of sine function)
= -sin(4π + 2π/3) (Dividing by 4π to obtain an angle within one revolution)
= -sin(2π/3) (Sine of an angle 2π/3 is known) = -√3/2.
tan x = tan(-14π/3) = tan(14π/3) (Using the symmetry of tangent function)
= tan(4π + 2π/3) (Dividing by 4π to obtain an angle within one revolution)
= tan(2π/3) (Tangent of an angle 2π/3 is known)= √3. sec x = sec(-14π/3) = sec(14π/3) (Using the symmetry of secant function)= sec(4π + 2π/3) (Dividing by 4π to obtain an angle within one revolution)= sec(2π/3) (Secant of an angle 2π/3 is known)= -2/√3. Therefore, the exact values in simplest form are: sin x = -√3/2, tan x = √3, sec x = -2/√3.
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Answer: The exact values of sin x, tan x and sec x in simplest form are: sin x = -√3/2, tan x = √3, sec x = -2/√3.
Step-by-step explanation: To solve this problem, we'll work with the angle x, which is equal to -14π/3. Part a: Determine the reference angle of x. The reference angle is the positive acute angle formed between the terminal side of the angle and the x-axis. To find the reference angle, we ignore the negative sign and convert the angle to its equivalent positive angle within one revolution.
The positive equivalent angle can be obtained by adding 2π (or 360 degrees) repeatedly until we obtain a positive angle. In this case, we have: -14π/3 + 2π = -14π/3 + 6π/3 = -8π/3. The reference angle of x is 8π/3. Part b: Find the exact values of sin x, tan x, and sec x in simplest form. sin x = sin(-14π/3) = -sin(14π/3) (Using the symmetry of sine function)
= -sin(4π + 2π/3) (Dividing by 4π to obtain an angle within one revolution)
= -sin(2π/3) (Sine of an angle 2π/3 is known) = -√3/2.
tan x = tan(-14π/3) = tan(14π/3) (Using the symmetry of tangent function)
= tan(4π + 2π/3) (Dividing by 4π to obtain an angle within one revolution)
= tan(2π/3) (Tangent of an angle 2π/3 is known)= √3. sec x = sec(-14π/3) = sec(14π/3) (Using the symmetry of secant function)= sec(4π + 2π/3) (Dividing by 4π to obtain an angle within one revolution)= sec(2π/3) (Secant of an angle 2π/3 is known)= -2/√3. Therefore, the exact values in simplest form are: sin x = -√3/2, tan x = √3, sec x = -2/√3.
Alinear trendline used to forecast sales for a given time period takes the form y = b+ bil. increases by , then the estimated y value all else e tone period, increases, b1; constant o tone period, increases, bo, constant bione period, increases; bo constant bi: one period, increases bi: constant
The linear trendline used to forecast sales for a given time period takes the form y = b0 + b1t, where y represents the estimated sales, b0 is the constant term, b1 is the coefficient of the time period variable (t), and t is the time period.
In this equation, the coefficient b1 determines the relationship between the time period and the estimated sales. If b1 increases, it means that for each additional time period, the estimated sales will also increase. On the other hand, if b1 is constant, it implies that the estimated sales do not change with each additional time period.
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Sami cut 6 and three fourth inches off a long roll of paper if the row 36 and one third inches long how long was the original roll of papers
Answer:
43 1 ÷12
Step-by-step explanation:
The computation of the length of the original roll of papers is shown below:
36 1 ÷3 and 6 3 ÷ 4 together
Now convert the above fractions into a number
36 1 ÷ 3 = 109 ÷3
And,
6 3 ÷4 = 27 ÷ 4
Now add these two numbers i.e.
109 ÷3 + 27 ÷ 4
= 436 + 81 ÷ 12
= 517 ÷12
= 43 1 ÷12
someone help please!! oh and pls don’t put any links
Answer:
10 visits
Step-by-step explanation:
So first let's make an equation, 7.74x + 8.93= 86.33. When you solve for x you get 10.
Please help if you wantbbrainleist! :(
Analyze and sketch a graph of the function. Find any intercepts, relative extrema, points of inflection, and asymptotes. (If an answer does not exist, enter DNE.)
f(x) = 3x2/3 − 2x
I keep getting stuck at taking the first derivative and solving for critical points. So please show all work and even some of the tedious algebra bits included so I can see where I'm messing up?
The function f(x) = 3x²/3 - 2x has no intercepts, a relative minimum at (1, -1), no points of inflection, and no asymptotes.
Intercepts: To find the x-intercepts, we set f(x) equal to zero and solve for x:
0 = 3x²/3 - 2x
0 = x² - 2x
0 = x(x - 2)
x = 0 or x = 2
Therefore, both x = 0 and x = 2 are not actual x-intercepts, but rather double roots.
Relative Extrema: To find the relative extrema, we take the derivative of f(x) and set it equal to zero,
f'(x) = 2x - 2
0 = 2x - 2
2 = 2x
x = 1
Substituting x = 1 back into the original function, we find f(1) = -1. Therefore, the relative minimum occurs at (1, -1).
f''(x) = 2
Since the second derivative is a constant, it never equals zero. Therefore, there are no points of inflection for this function.
Asymptotes: To determine if there are any asymptotes, we examine the behavior of the function as x approaches positive or negative infinity. Since the highest power of x in the function is 2, the graph does not approach any vertical asymptotes.
For horizontal asymptotes, we look at the limits as x approaches positive or negative infinity:
lim(x→∞) f(x) = lim(x→∞) (3x²/3 - 2x) = ∞
The limits approach positive infinity in both cases, indicating that there are no horizontal asymptotes. Graphically, the function represents a parabola that opens upwards, with a relative minimum at (1, -1).
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Which inequality is true if p = 3.4 ?
Answer:
the shape isnt angled thats what it means
Step-by-step explanation:
Subject:Mathematics
Answer:F : 13 1/2
Step-by-step explanation:27 divided by 2 is 13 1/2
Find the area of a sector with central angle 27/8 rad in a circle of radius 4 m.
The area of the sector with a central angle of 27/8 radians in a circle of radius 4 meters is 27 square meters.
To find the area of a sector, you can use the formula:
Area of Sector = (θ/2π) * πr²
where θ is the central angle in radians and r is the radius of the circle.
Given:
Central angle (θ) = 27/8 radians
Radius (r) = 4 meters
Substituting the given values into the formula, we have:
Area of Sector = (27/8 * 1/(2π)) * π * (4)^2
Simplifying the expression:
Area of Sector = (27/8 * 1/2) * (4)^2
Area of Sector = (27/16) * 16
Area of Sector = 27 square meters
Therefore, the area of the sector with a central angle of 27/8 radians in a circle of radius 4 meters is 27 square meters.
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Which equation represents a line which is perpendicular to the line
7x + 3y = -18?
Answer:
c y=6x+4
Step-by-step explanation:
What is the name of the blue dot located inside the curve of the parabola
below?
O A. Focus
B. Center
O C. Directrix
D. Vertex
The blue dot located inside the curve of the parabola is Focus.
What are Parts in Parabola?
The essential feature of a parabola is that all of its points are the same distance from a point called the focus and a line called the directrix. The vertex, axis, latus rectum, and focal length are also key components of a parabola.
The latus rectum, also known as the focal diameter, is the line segment that runs through the focus and parallel to the directrix. The focal diameter's endpoints are located on the curve.
A parabola is the set of all points ( x, y ) in a plane that are the same distance from a fixed line called the directrix and a fixed point not on the directrix (the focus).
A parabola will have three key components: a focus, a directrix, and a vertex. This upward-opening parabola demonstrates that all points,, along the parabola's curve will be the same distance from the focus and the directrix.
Thus, the blue dot located inside the curve of the parabola is Focus.
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PLEASE HELP!! I DON'T UNDERSTAND!!!!! I WILL MARK!!!!
Answer:
2
Step-by-step explanation:
first use order of operations.
1/2(4^2)-6
1/2*16-6
then simplify to get:
8-6=2
Answer: 2
Step-by-step explanation:
The question is asking you to substitute the b with 4 and find the solution to the expression.
[tex]\frac{1}{2}b^{2} -6[/tex]
[tex]=\frac{1}{2}(4)^{2} -6[/tex]
[tex]=\frac{1}{2}(16) -6[/tex] (PEMDAS so you do the exponent first)
[tex]=8 -6[/tex] (PEMDAS so you do the multiplication first)
[tex]=2[/tex]
how many square miles does ATC and radar services attempt to cover? How many aircraft at any given time is ATC monitoring, and spread over how many airports within the USA?
The work of providing air traffic control and radar services in the United States falls under the purview of the Federal Aviation Administration (FAA). ATC and radar services that cover all airspace over the United States, regardless of whether flights are domestic or international.
What is the radar servicesFAA provides ATC and radar services in the US. ATC and radar services cover all US airspace, including domestic and international flights. The FAA manages the NAS, covering 29.5 million sq mi. This includes airspace over the entire United States, including Alaska, Hawaii, Guam, Puerto Rico. VI ATC monitors varying number of aircraft based on time, weather, and traffic.
The FAA deals with 44k flights daily in the US. Note that this number may increase during peak travel periods. The FAA manages ATC for 13k+ US airports. Incl. international, regional, gen. aviation & priv. airstrips. The number can vary due to new or old airports.
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please help me .........
Answer: the answer is b
Step-by-step explanation:
Answer:
B
Step-by-step explanation:
first add them together. the y cancels out and your left with 3x=15. divide by 3 on both sides and you get x=5. The only answer with positive 5 as an x value is b
Roy received math test scores of 05, 90, 90, and 85.
7. What is Roy's median test score?
8. What score would Roy need to get on
his next test to have a mean of 92
Median = 87.5 Sorry about the other one
Let {N(t),t > 0} be a renewal process. Derive a renewal-type equation for E[SN (1)+1).
The renewal-type equation for E[SN(1)+1] is E[SN(1)+1] = 2, indicating that the expected value of the sum of the number of renewals by time 1 plus 1 is equal to 2.
To derive a renewal-type equation for E[SN(1)+1], we can use the renewal-reward theorem.
Let Tn be the interarrival times of the renewal process, where n represents the nth renewal. The random variable N(t) represents the number of renewals that occur by time t.
Using the renewal-reward theorem, we have:
E[SN(1)+1] = E[T1 + T2 + ... + TN(1) + 1]
Since the interarrival times are independent and identically distributed (i.i.d.), we can express this as:
E[SN(1)+1] = E[T] * E[N(1)] + 1
Now, we need to compute the expressions for E[T] and E[N(1)].
E[T] represents the expected interarrival time, which is equal to the reciprocal of the renewal rate. Let λ be the renewal rate, then E[T] = 1/λ.
E[N(1)] represents the expected number of renewals by time 1. This can be calculated using the renewal equation:
E[N(t)] = λ * t
Therefore, E[N(1)] = λ * 1 = λ.
Substituting these expressions back into the renewal-type equation, we have:
E[SN(1)+1] = (1/λ) * λ + 1 = 1 + 1 = 2
Hence, the renewal-type equation for E[SN(1)+1] is E[SN(1)+1] = 2.
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The integrat (cos(x - 3) dx is transformed into L' (t)dt by applying an appropriate change of variable, then g(t) i g(t) = 1/2 cos (t-3/2) This option g(t) = 1/2 sin(t-3/2)
The integrat (cos(x - 3) dx is transformed into L' (t)dt an g(t) = 1/2 sin(t-3/2) is incorrect. The correct option g(t) = 1/2 cos(t-3/2).
To transform the integral ∫cos(x - 3)dx into L'(t)dt using an appropriate change of variable, the substitution method make the substitution:
t = x - 3
To find dt, differentiate both sides of the equation with respect to x:
dt = dx
substitute these expressions into the integral:
∫cos(x - 3)dx = ∫cos(t)dt
The integral has been transformed into ∫cos(t)dt.
regarding the options for g(t),
Option 1: g(t) = 1/2 cos(t-3/2)
Taking the derivative of g(t) with respect to t,
g'(t) = -(1/2)sin(t - 3/2)
Option 2: g(t) = 1/2 sin(t-3/2)
Taking the derivative of g(t) with respect to t,
g'(t) = (1/2)cos(t - 3/2)
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I’ll mark you brainlieist
I need help with short sides of the triangles on Pythagorean theorem
Answer:
5
Step-by-step explanation:
13² - 12² = 25
√25 = 5
Have a great day <3
Osing Trig to Find a Side Apr 06, 5:40:44 PM In AOPQ, the measure of ZQ=90°, the measure of Z0=26°, and QO = 4.9 feet. Find the length of PQ to the nearest tenth of a foot. P (hypotenuse) X (opp. of 20) 2009 Q 4.9
Answer:5.4
Step-by-step explanation:
Show the enteiws to close $2000 in expense, $5000 in revenue, and
$500 in dividens.
To close the $2,000 in expenses, $5,000 in revenue, and $500 in dividends, we need to transfer these amounts to the appropriate accounts and close the temporary accounts at the end of the accounting period. Here are the journal entries to close these amounts:
Close Expenses:
Date | Account | Debit | Credit
End of Year | Expenses | $2,000 |
| Income Summary | | $2,000
Close Revenue:
Date | Account | Debit | Credit
End of Year | Income Summary | $5,000 |
| Revenue | | $5,000
Close Dividends:
Date | Account | Debit | Credit
End of Year | Retained Earnings | $500 |
| Dividends | | $500
After these closing entries, the balances of the temporary accounts (Expenses, Revenue, and Dividends) will be zero, and their respective amounts will be transferred to the Income Summary and Retained Earnings accounts. The Income Summary account will show the net income (revenue minus expenses) for the period.
Please note that the specific account titles may vary depending on the company's chart of accounts, so make sure to use the appropriate account titles according to your specific chart of accounts.
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A scalene triangle has sides measuring 200 feet, 107 feet, and 221 feet. What is the
perimeter of the triangle?
Answer:
P = 528
Step-by-step explanation:
P = a + b + c = 200 + 107 + 221 = 528
A pool manager balances the pH level of a pool. The price of a bucket of chlorine tablets is $90, and the price of a pH test kit is $11. The manager uses a coupon that applies a 40% discount to the total cost of the two items. How much money does the pool manager pay for each item?
Answer:
The chlorine tablets are $36 and the pH test kit is $4.40.
Step-by-step explanation:
So what I first did was add 90+11 (because $90 for the chlorine tablets and $11 for the pH kit). The answer you should get is 101. From their I did 101 multiplied by 40% or 101 multiplied by .40 (40% and 0.40 are the same thing). You now get $40.4.
Now I did 90·40%. I got $36
Then I did 11·40%. I got 4.4
I added them up and got $40.4 which was my new price so now we know that the chlorine tablets are $36 and the pH test kit is $4.40.
Which point is a solution to the inequality in this graph
Given:
The graph of an inequality.
To find:
The point which is a solution of the given graph of inequality.
Solution:
From the given graph it is clear that the boundary line of the graph is a dotted line. It means the points lie in shaded region are in the solution set but the points on the line are not included in the solution set.
The points (3,2) and (-3,-6) are lie on the boundary line. it means they are not the solution of the inequality represented by the given graph.
Point (5,0) lies on the positive x-axis at the distance of 5 units from the origin and it doesn't lies in the shaded region. So, (5,0) is not a solution.
Point (0,5) lies on the negative y-axis at the distance of 5 units from the origin and it lies in the shaded region. So, (0,5) is a solution.
Therefore, the correct option is B.
The scores on the Wechsler Adult Intelligence Scale are approximately Normal, with 100 and 11. The proportion of adults with scores above 110 is closest to 0.25 b.0.33. c0.14. 4.0.18 Colleges often rely heavily on raising money for an "annual fund" to support operations. Alumni are typically solicited for donations to the annual fund. Studies suggest that the graduate's smal income is a good predicar of the amount of money he or she would be willing to donate, and there is a reasonably strong, positive, linear relationship between these variables. In the stadies described a annual income is an explanatory variable. b the correlation between annual income and the size of the donation is positive. c the size of the donation to the annual fund is the response variable. d. All of the answer options are correct.
The proportion of adults with scores above 110 is 0.1841.
Here, we have,
It should be noted that from the information illustrated, Wechsler Adult Intelligence Scale scores are approximately Normal, with a mean of 100 and a standard deviation of 11.
The formula to use will be:
P(a < Z < b) = P(Z < b) – P(Z < a)
a = lower value
b = higher value
Z = z value
It should be noted that the proportion of adults with scores above 110 will be:
= P(x > 110)
= P(z > 110 - 100/11)
= P(z > 0.90)
= 1 - 0.8159
= 0.1841
Therefore, this illustrates those that has scores of more than 110.
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The volume of a rectangular prism is 1,560 cm3. The height is 12 cm. The width is w and the length is w + 3. Find w.
Answer:
w=10 cm
Step-by-step explanation:
The formula for the volume of a rectangular prism is V=whl.
So in this case, the equation would be 1560=w·12·(w+3). Then, we can simplify this equation.
1. Divide 12 to both sides of the equation. 1560/12=130. The equation becomes 130=w(w+3).
2. Distribute w through the parentheses, the equation becomes 130=w²+3w.
3. Then, -130 from both sides of the equation, so we can get the quadratic: w²+3w-130=0.
4. Factor the quadratic. w²+3w-130=(w-10)(w+13).
5. (w-10)(w+13)=0. w=10 or w=-13. However, w is the width of a rectangular prism, can the width of a shape be negative? No. So we can ignore the solution w=-13. Therefore, w=10cm.
6. To make sure our answer is correct, let's substitute the values back into the volume of a rectangular prism formula: V=whl. w=10; h=12; l=10+3=13; V=10(12)(13)=120*13=1560 cm³. As a result, our answer is correct, w=10 cm.
Hope this helps, have a nice day.