Domain definition in math

The answer is D. x = 32 and x = 0

============================================================

Explanation:

Think of x-16 as y. In other words, let y = x-16

We have (x-16)^2 = 256 become y^2 = 256.

Solve for y to get y = 16 or y = -16. You apply the square root to both sides. Don't forget about the plus/minus. This is because (-16)^2 = (-16)*(-16) = 256. We square the negative as well.

--------

If y = 16, then y = x-16 solves for x to get

y = x-16

16 = x-16

x-16 = 16

x = 16+16

x = 32 is one solution

---------

Plug in y = -16 and isolate x

y = x-16

x-16 = y

x-16 = -16

x = -16+16

x = 0 is the other solution

----------

We can check each answer by plugging them back into the original equation

Let's try x = 0

(x-16)^2 = 256

(0-16)^2 = 256

(-16)^2 = 256

256 = 256 ... works

Now try x = 32

(x-16)^2 = 256

(32-16)^2 = 256

(16)^2 = 256

256 = 256 ... also works; both answers have been confirmed

Answer:

The formula for calculating the probability of two independent events is the product of the probabilities of each event occurring. In other words, if event A and event B are independent, then the probability of both events occurring is calculated as P(A and B) = P(A) * P(B).

The power of a that makes it equal to b is x = 5/6

The square root of a number is a number when multiplied by itself gives another number.

The cube root of a number is a number when multiplied by itself three times gives another number.

The power of a number is the exponent of that number

Since two numbers, a and b, are each greater than zero, and 4 times the square root of a is equal to 9 times the cube root of b. It follws that

4√a = 9∛b

⇒∛b = 4√a/9

⇒ b = ∛[4√a/9]

Since we require the  of x is a, to the x power equal to b if a = 2/3.

This implies that

aˣ = b

Substituting b into the equation, we have

aˣ =  ∛[4√a/9]

Since a = 2/3, we have

(2/3)ˣ =  ∛[4√(2/3)/9]

(2/3)ˣ =  ∛[4/9 × √(2/3)]

(2/3)ˣ =  ∛[2²/3² × √(2/3)]

(2/3)ˣ =  ∛[(2/3)² × √(2/3)]

(2/3)ˣ =  ∛[2/3)

\((\frac{2}{3} )^{x} = \sqrt[3]{(\frac{2}{3} )^{2 + \frac{1}{2}} } \\(\frac{2}{3} )^{x} = \sqrt[3]{(\frac{2}{3} )^{\frac{5}{2}} } \\(\frac{2}{3} )^{x} = (\frac{2}{3} )^{\frac{5}{2} \times \frac{1}{3} } \\(\frac{2}{3} )^{x} = (\frac{2}{3} )^{\frac{5}{6}\)

Equating both powers, we see that x = 5/6

So, the power of a is x = 5/6

Learn more about powers of exponents here:

https://brainly.com/question/26130878

#SPJ1

The congruency relationship which hold true after transformation is

YZ ≅ AB

What is congruent ?

In mathematics, the term "congruent" refers to figures and shapes that can be flipped or rearranged to match up with other ones. These forms can be mirrored to produce related shapes.

Explanation:

A translation is known as a rigid transformation.  This is because it preserves congruence.  Since CAB is created from XYZ, this tells us that:

X corresponds to C; Y corresponds to A; Z corresponds to B; XY corresponds to CA; YZ corresponds to AB; and XZ corresponds to CB.

Since the triangles are congruent, the corresponding pieces are congruent as well.

Out of these corresponding congruent pieces of triangles, the only one in our options is YZ ≅ AB.

Hence the answer is YZ≅AB.

To learn more about congruent click on the link below

https://brainly.com/question/1675117

#SPJ4

6) The solution is:  x = 1, y = -4, z = -3

7) The solution is: x = 1, y = 3, z = 5/7

8) The solution to the system of equations is: x = -11, y = 0, z = -5.

A quadratic equation is a polynomial equation of the second degree, which means it has a degree of two.

6.) To solve the systems of equations, we can use the Gaussian elimination method.

We will rewrite this system in the augmented matrix form:

[ 3 -1 2 | -4 ]

[ 6 -2 4 | -8 ]

[ 2 -1 3 | -10]

[ 3 -1 2 | -4 ]

[ 0 0 -4/3 | 4 ]

[ 0 1/3 7/3 | -14/3]

Then, we can add (1/3) times the second row to the third row:

[ 3 -1 2 | -4 ]

[ 0 0 -4/3 | 4 ]

[ 0 1/3 21/9 | -10/3]

[ 3 -1 2 | -4 ]

[ 0 0 -4/3 | 4 ]

[ 0 3 7 | -30 ]

7.) We can rewrite this system in the augmented matrix form:

[ 1 1 -1 | 4 ]

[ 3 2 4 | -17 ]

[-1 5 1 | 8 ]

[ 1 1 -1 | 4 ]

[ 0 -1 7 | -29 ]

[ 0 6 0 | 12 ]

Then, we can multiply the second row by -1 and add it to the third row:

[ 1 1 -1 | 4 ]

[ 0 -1 7 | -29 ]

[ 0 0 -42 | -150]

8.) To solve for x, y, and z in the system of equations:

x + 5y - 2z = -1 (equation 1)

-x - 2y + z = 6 (equation 2)

-2x - 7y + 3z = 7 (equation 3)

(1) + (2): 3y - z = 5

2(1) + (3): 13y - z = 5

3y - z = 5 (equation A)

13y - z = 5 (equation B)

10y = 0

3(0) - z = 5

z = -5

x + 5(0) - 2(-5) = -1

x + 10 = -1

x = -11

To know more about matrix visit:

https://brainly.com/question/27572352

#SPJ1

The solution to the equation \((\sum\limits^{\infty}_{i=1} \frac{1}{2^i}) + (\sum\limits^{\infty}_{i=1} \frac{9}{10^i}) + \sin^2(\theta) + \cos^2(\theta) + e^2\) is 10.3891

From the question, we have the following parameters that can be used in our computation:

\((\sum\limits^{\infty}_{i=1} \frac{1}{2^i}) + (\sum\limits^{\infty}_{i=1} \frac{9}{10^i}) + \sin^2(\theta) + \cos^2(\theta) + e^2\)

Using the following trigonometry ratio

sin²(x) + cos²(x) = 1

We have

\((\sum\limits^{\infty}_{i=1} \frac{1}{2^i}) + (\sum\limits^{\infty}_{i=1} \frac{9}{10^i}) + \sin^2(\theta) + \cos^2(\theta) + e^2 = (\sum\limits^{\infty}_{i=1} \frac{1}{2^i}) + (\sum\limits^{\infty}_{i=1} \frac{9}{10^i}) + 1 + e^2\)

The sum to infinity of a geometric series is

S = a/(1 - r)

So, we have

\((\sum\limits^{\infty}_{i=1} \frac{1}{2^i}) + (\sum\limits^{\infty}_{i=1} \frac{9}{10^i}) + \sin^2(\theta) + \cos^2(\theta) + e^2 = \frac{1/2}{1 - 1/2} + \frac{9/10}{1 - 1/10} + 1 + e^2\)

So, we have

\((\sum\limits^{\infty}_{i=1} \frac{1}{2^i}) + (\sum\limits^{\infty}_{i=1} \frac{9}{10^i}) + \sin^2(\theta) + \cos^2(\theta) + e^2 = 1 + 1 + 1 + e^2\)

Evaluate the sum

\((\sum\limits^{\infty}_{i=1} \frac{1}{2^i}) + (\sum\limits^{\infty}_{i=1} \frac{9}{10^i}) + \sin^2(\theta) + \cos^2(\theta) + e^2 = 3 + e^2\)

This gives

\((\sum\limits^{\infty}_{i=1} \frac{1}{2^i}) + (\sum\limits^{\infty}_{i=1} \frac{9}{10^i}) + \sin^2(\theta) + \cos^2(\theta) + e^2 = 10.3891\)

Hence, the solution to the equation is 10.3891

Read more about summation notation at

brainly.com/question/15973233

#SPJ1

Answer:

1.92

Step-by-step explanation:

2.40 is more than 1.92

The paired t-test statistic is infinity, which indicates a significant difference in the means of math scores and science scores.

Let's consider an example of a paired t-test and a two-sample t-test on a dataset of 5 students' scores in two different exams: math and science. Let's assume that the students took both exams, and their scores are paired. We want to test whether there is a significant difference in the means of math scores and science scores.

We calculate the difference between each student's math score and science score and compute the mean difference and standard deviation of the differences:

Student Math Score Science Score Difference

1                      80      75                        5

2                      90      85                        5

3                      85      80                        5

4                       95      90                        5

5                    75              70                        5

Mean difference = 5

Standard deviation of differences = 0

We can calculate the paired t-test statistic as:

t = (mean difference - hypothesized difference) / (standard deviation of differences / square root of sample size)

Let's assume the hypothesized difference is 0 (i.e., there is no difference between the means). Then the t-test statistic is:

t = (5 - 0) / (0 / √(10)) = infinity

To know more about t-test here.

https://brainly.com/question/15870238

#SPJ4

Answer:

You need to be specific on which is the length, height and width. Making the most basic assumption I would have to assume the answer is 228

Step-by-step explanation:

Formula for the surface area of a rectangular prism is A=2(wl+hl+hw)

A = 2 x (9 x 6 + 4 x 6 + 4 x 9)

A = 2 x (54 + 24 + 36)

A = 2 (114)

A = 228

Answer:

an obtuse angle is an angle that is wider than 90⁰, 100⁰ for example is an obtuse angle

The parent function of a quadratic equation is f(x) = x². The function that is transformed to the right and down from the parent function with a minimum is given by f(x) = a(x - h)² + k.

The equation has the same shape as the parent quadratic function. However, it is shifted up, down, left, or right, depending on the values of  a, h, and k.

For a parabola to have a minimum value, the value of a must be positive. If a is negative, the parabola will have a maximum value.To find the vertex of the parabola in this form, we use the vertex form of a quadratic equation:f(x) = a(x - h)² + k, where(h, k) is the vertex of the parabola.The vertex is the point where the parabola changes direction. It is the minimum or maximum point of the parabola. In this case, the parabola is transformed to the right and down from the parent function, f(x) = x². Therefore, h > 0 and k < 0.

Know more about quadratic equation here:

https://brainly.com/question/12186700

#SPJ11

The shortest distance to reach the starting point by Dolores is 2.7 kilometres.

The shortest distance will be the line directly connecting the two points. This will form the right angled triangle and hence can directly be calculated. We will be calculating hypotenuse from Pythagoras theorem.

Distance² = 2.2² + 1.5²

Taking square on Right Hand Side

Distance² = 4.84 + 2.25

Performing addition on Right Hand Side of the equation

Distance² = 7.09

Taking square root on Right Hand Side

Distance = ✓7.09

Distance = 2.7 kilometres

Thus, the shortest distance back to the starting point is 2.7 kilometres.

Learn more about Pythagoras theorem -

https://brainly.com/question/343682

#SPJ1

Answer:

y - 4= -3(x - 2)

I'm 101% sure

Answer:

11

Step-by-step explanation:

the curved lone starts at 0 and ends in 11

Step-by-step explanation:

If 12x = -8 (10 - x) then x = ? F . 20 G . 8 H .7 3/ 11 j .6 2 / 13 k .-20

Answer:

whither what I got npthing

Answer:

x = 1, y = 3

Step-by-step explanation:

use substitution method since the second equation is already solved for 'x'

substitute '2y-5' for 'x' in the first equation:

2y - 5 + 3y = 10

5y - 5 = 10

5y = 15

y = 3

now substitute 3 for y to find out what x equals:

x + 3(3) = 10

x + 9 = 10

x = 1

a) The deceleration can be found using the formula:

deceleration = (final velocity - initial velocity) / time

When slamming on the brakes, the final velocity is 0 mph, the initial velocity is 75 mph, and the time taken to stop is unknown. So, the algebraic expression for deceleration is:

deceleration = (0 - 75) / time

b) The stopping distance can be calculated using the formula:

stopping distance = (initial velocity)^2 / (2 * deceleration)

Substituting the given values of initial velocity and deceleration, we get:

stopping distance = (75)^2 / (2 * deceleration) = (75)^2 / (2 * (0 - 75 / time))

Simplifying this expression, we get:

stopping distance = 138 * 75 / 60 = 172.5 feet

So, the skid marks left are 172.5 feet.

c) The time taken to stop can be found by dividing the stopping distance by the initial velocity and then multiplying by 3600 to convert from hours to seconds. Using the same values as in part b), we get:

time taken to stop = stopping distance / initial velocity * 3600 = 172.5 / 75 * 3600 = 24.2 seconds

So, it took approximately 24.2 seconds to stop.

Answer:

0.0031

Step-by-step explanation:

31/10000=0.0031

you divided by 10000 and it has four zeroes so you should have four numbers after 0

Answer:

.0031

Step-by-step explanation:

The 10,000ths place is the fourth place to the right of the decimal. This is where you place the 1 in 31, as it is out of 10,000, and the 3 goes to the left of it, so it is .0031.

To find the joint density function, we need to calculate the Jacobian determinant of the transformation from (x, y) to (u, v)

The joint distribution of (u, v), where u and v are defined as

\(u = \frac{x}{{\sqrt{x^2 + y^2}}}\)  and \(v = \frac{y}{{\sqrt{x^2 + y^2}}}\), is given by:

\(f_{U,V}(u,v) = \frac{1}{{2\pi}} \cdot e^{-\frac{1}{2}(u^2 + v^2)}\)

To find the joint density function, we need to calculate the Jacobian determinant of the transformation from (x, y) to (u, v):

\(J = \frac{{du}}{{dx}} \frac{{du}}{{dy}}\)

\(\frac{{dv}}{{dx}} \frac{{dv}}{{dy}}\)

Substituting u and v in terms of x and y, we can evaluate the partial derivatives:

\(\frac{{du}}{{dx}} &= \frac{{y}}{{(x^2 + y^2)^{3/2}}} \\\frac{{du}}{{dy}} &= -\frac{{x}}{{(x^2 + y^2)^{3/2}}} \\\frac{{dv}}{{dx}} &= -\frac{{x}}{{(x^2 + y^2)^{3/2}}} \\\frac{{dv}}{{dy}} &= \frac{{y}}{{(x^2 + y^2)^{3/2}}}\)

Therefore, the Jacobian determinant is:

\(J &= \frac{y}{{(x^2 + y^2)^{\frac{3}{2}}}} - \frac{x}{{(x^2 + y^2)^{\frac{3}{2}}}} \\&= -\frac{x}{{(x^2 + y^2)^{\frac{3}{2}}}} + \frac{y}{{(x^2 + y^2)^{\frac{3}{2}}}} \\J &= \frac{1}{{(x^2 + y^2)^{\frac{1}{2}}}}\)

Now, we can find the joint density function of (u, v) as follows:

\(f_{U,V}(u,v) &= f_{X,Y}(x,y) \cdot \left|\frac{{dx,dy}}{{du,dv}}\right| \\&= f_{X,Y}(x,y) / J \\&= f_{X,Y}(x,y) \cdot (x^2 + y^2)^{\frac{1}{2}}\)

Substituting the standard normal density function

\(f_{X,Y}(x,y) &= \frac{1}{2\pi} \cdot e^{-\frac{1}{2}(x^2 + y^2)} \\f_{U,V}(u,v) &= \frac{1}{2\pi} \cdot e^{-\frac{1}{2}(x^2 + y^2)} \cdot (x^2 + y^2)^{\frac{1}{2}} \\&= \frac{1}{2\pi} \cdot e^{-\frac{1}{2}(u^2 + v^2)}\)

Therefore, the joint distribution of (u, v) is given by:

\(f_{U,V}(u,v) &= \frac{1}{2\pi} \cdot \exp\left(-\frac{1}{2}(u^2 + v^2)\right)\)

Learn more about joint probability distributions:

https://brainly.com/question/32099581

#SPJ11

To find the joint density function, we need to calculate the Jacobian determinant of the transformation from (x, y) to (u, v)

The joint distribution of (u, v) is a bivariate normal distribution with mean (0,0) and variance-covariance matrix

\(\begin{bmatrix}2 & 0 \0 & 2\end{bmatrix}\)

The joint distribution of (u, v) can be found by transforming the independent random variables x and y using the following formulas:

\( u = x + y\)
\( v = x - y \)

To find the joint distribution of (u, v), we need to find the joint probability density function (pdf) of u and v.

Let's start by finding the Jacobian determinant of the transformation:

\(J = \frac{{\partial (x, y)}}{{\partial (u, v)}}\)

\(= \frac{{\partial x}}{{\partial u}} \cdot \frac{{\partial y}}{{\partial v}} - \frac{{\partial x}}{{\partial v}} \cdot \frac{{\partial y}}{{\partial u}}\)

\(= \left(\frac{1}{2}\right) \cdot \left(-\frac{1}{2}\right) - \left(\frac{1}{2}\right) \cdot \left(\frac{1}{2}\right)\)

\(J = -\frac{1}{2}\)

Next, we need to express x and y in terms of u and v:

\(x = \frac{u + v}{2}\)

\(y = \frac{u - v}{2}\)

Now, we can find the joint pdf of u and v by substituting the expressions for x and y into the joint pdf of x and y:

\(f(u, v) = f(x, y) \cdot |J|\)

\(f(u, v) = \left(\frac{1}{\sqrt{2\pi}}\right) \cdot \exp\left(-\frac{x^2}{2}\right) \cdot \left(\frac{1}{\sqrt{2\pi}}\right) \cdot \exp\left(-\frac{y^2}{2}\right) \cdot \left|-\frac{1}{2}\right|\)

\(f(u, v) = \frac{1}{2\pi} \cdot \exp\left(-\frac{u^2 + v^2}{8}\right)\)

Therefore, the joint distribution of (u, v) is given by:

\(f(u, v) = \frac{1}{2\pi} \cdot \exp\left(-\frac{{u^2 + v^2}}{8}\right)\)

In summary, the joint distribution of (u, v) is a bivariate normal distribution with mean (0,0) and variance-covariance matrix

\(\begin{bmatrix}2 & 0 \0 & 2\end{bmatrix}\)

Learn more about joint probability distributions:

brainly.com/question/32099581

#SPJ11

Answer:

b 2，3，8

Step-by-step explanation:

Y=234678

Z=278910

X=12358

234678910 n 12358

=238

Answer:

3 miles

Step-by-step explanation:

Slope is simply the change in position per unit rate. We could conclude that the steepness of the Applachian is the rate of change in position per unit time. Since the distance hiked on the Applachian is said to be 3 miles per hour, then we can conclude that the slope of the applachian is 3 miles

This is the change in miles / change in time.

Chang in miles = 3

Change in time = 1

Slope = 3/1

Slope = 3

Answer:

B i took the test/quiz

Answer:

C

Step-by-step explanation:

If I helped can you mark me brainliest? :)

Therefore, the compound interest of the annuity due of BD 400 paid each 4 months for 6.2 years at a nominal rate of 3% per annum is BD 40,652.17.

Compound interest of an annuity due can be calculated using the formula:A = R * [(1 + i)ⁿ - 1] / i * (1 + i)

whereA = future value of the annuity dueR = regular paymenti = interest raten = number of payments First, we need to calculate the effective rate of interest per period since the nominal rate is given per annum. The effective rate of interest per period is calculated as

:(1 + i/n)^n - 1 = 3/1003/100 = (1 + i/4)^4 - 1

(1 + i/4)^4 = 1.0075i/4 = (1.0075)^(1/4) - 1i = 0.0303So,

the effective rate of interest per 4 months is 3.03%.Next, we can substitute the given values in the formula:

A = BD 400 * [(1 + 0.0303)^(6.2 * 3) - 1] / 0.0303 * (1 + 0.0303)A = BD 400 * [4.227 - 1] / 0.0303 * 1.0303A = BD 400 * 101.63A = BD 40,652.17

For such more question on interest

https://brainly.com/question/25720319

#SPJ8

Answer:

im sorry i don't understand!

Step-by-step explanation:

Full question required!

Answer:

i think its b or C

Step-by-step explanation:

Answer:

1.5

Step-by-step explanation:

3(8 – 4x) < 6(x – 5)?

1

1

-5

WO

+

4

-4

-3

-2

-1

0

1

2

5

+

-1

+

1

+

2

-5

-4

-3

-2

0

3

4

5

-4

-3

+

1

-5

+

2

-1

+

3

-2

0

4

5

-+

-5

+

-1

-3

+

2

-4

-2

0

1

3

4

5

This doesn't make any sense

Answer:

9/10

Step-by-step explanation:

Find a common denominator

4/5 × 2 = 8/10

8/10 + 1/10 = 9/10

Hey there!

4/5 + 1/10

= 4 × 2 / 5 × 2 + 1/10

= 8/10 + 1/10

= 8 + 1 / 10 + 0

= 9/10

Therefore, your answer should be:

9/10

Good luck on your assignment \& enjoy your day!

~Amphitrite1040:)

The total electric force on the 7.00−μC charge is 0.872 N at an angle of 330°.

The force exerted on the 7.00−μC charge by the 2.00−μC charge is

F₁​ ​= \(k_{e}\)q₁q₂​​/r₂ r^

= [(8.99×10⁹N.m²/C²)(7.00×10^−6C)(2.00×10^−6C) ×(cos60°i^+sin60°j^​)]/ (0.500m)²

F₁ ​= (0.252i^+0.436j^​)N

Similarly, the force on the 7.00μC charge by the −4.00−μC charge is

F₂​ ​= \(k_{e}\)q₁​q₃/​r² ​​r^

= [(8.99×10⁹N.m²/C²)(7.00×10^−6C)(−4.00×10^−6C)​×(cos60°i^−sin60°j^​)]/(0.500m)²

F₂​ ​=(0.503i^−0.872j^​)N

Hence, the total force on the charge 7.00−μC  is

F = F₁​​+F₂​

​=(0.755i^−0.436j^​)N

We can also note the total force as:

F = (0.755N)i^−(0.436N)j^​ = 0.872N

To know more about electric force, here

https://brainly.com/question/29141236

#SPJ4

--This question is incomplete; the complete question is

"Three charged particles are located at the corners of an equilateral triangle, as shown in Figure. Calculate the total electric force on the 7.00−μC charge."--

The corrected statement of the Pumping Lemma for regular languages will be: Let L be a regular language. Then ∃n ≥ 0 such that for any w ∈ L such that |w| > n, ∃x, y, z such that w = xyz where: |xy| ≤ n, x ≠ λ, |y| > 0 and xyiz ∈ L for all i ≥ 0.

∃n≥1 or any w∈L such that ∣w∣>n,∃x,y,z such that w=xyz where: ∣xy∣≤n,  x ≠ λ and xyiz∈L for all i≥0. Here are the three errors in the given statement of Pumping Lemma for regular languages:

i. The statement should be xyiz where i ≥ 0, instead of xyiz for all i ≥ 1.

ii. The value of n is given to be ≥ 1 but it should be ≥ 0.

iii. There should be a condition that the length of y (i.e. |y|) should be > 0.

You can learn more about Pumping Lemma at: brainly.com/question/33347569

#SPJ11

The velocity, acceleration, and speed of the particle are
v(t) = 2ti + 2j + (7/t)k, a(t) = 2i - (7/t^2)k, s(t) = √((4t^4 + 4t^2 + 49) / t^2) respectivelty.

To find the velocity, acceleration, and speed of a particle with the given position function, we can differentiate the position function with respect to time. Let's go step by step:

1. Position function:
The given position function is r(t) = t^2i + 2tj + 7ln(t)k.

2. Velocity:
To find the velocity, we need to differentiate the position function with respect to time.
Velocity (v) is the derivative of position (r) with respect to time (t).

Taking the derivative of each component of the position function:
d/dt (t^2) = 2t
d/dt (2t) = 2
d/dt (7ln(t)) = 7/t

So, the velocity function (v) is:
v(t) = 2ti + 2j + (7/t)k

3. Acceleration:
To find the acceleration, we need to differentiate the velocity function with respect to time.
Acceleration (a) is the derivative of velocity (v) with respect to time (t).

Taking the derivative of each component of the velocity function:
d/dt (2t) = 2
d/dt (2) = 0
d/dt (7/t) = -7/t^2

So, the acceleration function (a) is:
a(t) = 2i + 0j - (7/t^2)k
or simply,
a(t) = 2i - (7/t^2)k

4. Speed:
The speed of a particle is the magnitude of its velocity vector.
The magnitude of a vector is found using the Pythagorean theorem.
For the velocity function v(t) = 2ti + 2j + (7/t)k, the magnitude of v(t) is:

|v(t)| = √((2t)^2 + (2)^2 + (7/t)^2)
      = √(4t^2 + 4 + 49/t^2)
      = √((4t^4 + 4t^2 + 49) / t^2)

Therefore, the speed function (s) is:
s(t) = √((4t^4 + 4t^2 + 49) / t^2)

This is the velocity, acceleration, and speed of the particle with the given position function.

Learn more about  velocity, acceleration, and speed https://brainly.com/question/25749514

#SPJ11

1   Y is distributed as N(aμ + b, a^2σ^2), as desired.

2  We have shown that under these conditions, E[XY] = E[X]E[Y].

To prove that Y is distributed as N(aμ + b, a^2σ^2), we need to show that the mean and variance of Y match those of a normal distribution with parameters aμ + b and a^2σ^2, respectively.

First, let's find the mean of Y:

E(Y) = E(aX + b) = aE(X) + b = aμ + b

Next, let's find the variance of Y:

Var(Y) = Var(aX + b) = a^2Var(X) = a^2σ^2

Therefore, Y is distributed as N(aμ + b, a^2σ^2), as desired.

We can use the definition of covariance to prove that E[XY] = E[X]E[Y]. By the properties of expected value, we know that:

E[XY] = ∫∫ xy f(x,y) dxdy

where f(x,y) is the joint probability density function of X and Y.

Then, we can use the fact that X and Y are independent to simplify the expression:

E[XY] = ∫∫ xy f(x) f(y) dxdy

= ∫ x f(x) dx ∫ y f(y) dy

= E[X]E[Y]

where f(x) and f(y) are the marginal probability density functions of X and Y, respectively.

Therefore, we have shown that under these conditions, E[XY] = E[X]E[Y].

Learn more about distributed here:

https://brainly.com/question/29664127

#SPJ11