Slope intercept form 8x-6y=18

Suppose that Y has density function

zvonat [6]

I'm assuming

$f(y)=\begin{cases}ky(1-y)&\text{for }0\le y\le1\\0&\text{otherwise}\end{cases}$

(a) f(x) is a valid probability density function if its integral over the support is 1:

$\displaystyle\int_{-\infty}^\infty f(x)\,\mathrm dx=k\int_0^1 y(1-y)\,\mathrm dy=k\int_0^1(y-y^2)\,\mathrm dy=1$

Compute the integral:

$\displaystyle\int_0^1(y-y^2)\,\mathrm dy=\left(\frac{y^2}2-\frac{y^3}3\right)\bigg|_0^1=\frac12-\frac13=\frac16$

So we have

k / 6 = 1 → k = 6

(b) By definition of conditional probability,

P(Y ≤ 0.4 | Y ≤ 0.8) = P(Y ≤ 0.4 and Y ≤ 0.8) / P(Y ≤ 0.8)

P(Y ≤ 0.4 | Y ≤ 0.8) = P(Y ≤ 0.4) / P(Y ≤ 0.8)

It makes sense to derive the cumulative distribution function (CDF) for the rest of the problem, since F(y) = P(Y ≤ y).

We have

$\displaystyle F(y)=\int_{-\infty}^y f(t)\,\mathrm dt=\int_0^y6t(1-t)\,\mathrm dt=\begin{cases}0&\text{for }y$

Then

P(Y ≤ 0.4) = F (0.4) = 0.352

P(Y ≤ 0.8) = F (0.8) = 0.896

and so

P(Y ≤ 0.4 | Y ≤ 0.8) = 0.352 / 0.896 ≈ 0.393

(c) The 0.95 quantile is the value φ such that

P(Y ≤ φ) = 0.95

In terms of the integral definition of the CDF, we have solve for φ such that

$\displaystyle\int_{-\infty}^\varphi f(y)\,\mathrm dy=0.95$

We have

$\displaystyle\int_{-\infty}^\varphi f(y)\,\mathrm dy=\int_0^\varphi 6y(1-y)\,\mathrm dy=(3y^2-2y^3)\bigg|_0^\varphi = 0.95$

which reduces to the cubic

3φ² - 2φ³ = 0.95

Use a calculator to solve this and find that φ ≈ 0.865.

8 0

3 years ago

Find approximate solution for the equation in the interval -π tan x= 2 csc x

tresset_1 [31]

Answer:

The angle in the 1st quadrant is 1.144 and in the 4th quadrant is -1.144

∴ The answer is (a)

Step-by-step explanation:

* The domain of the function is -π < x < π

- Then the angle is in the 1st or 4th quadrant

∵ tan(x) = 2 csc(x)

∵ tan(x) = sin(x)/cos(x)

∵ csc(x) = 1/sin(x)

∴ sin(x)/cos(x) = 2(1/sin(x)) = 2/sin(x) ⇒ using cross multiplication

∴ sin²(x) = 2cos(x)

∵ sin²(x) = 1 - cos²(x) ⇒ substitute it in the last step

∴ 1 - cos²(x) = 2cos(x) ⇒ arrange the terms in one side

∴ cos²(x) + 2cos(x) - 1 = 0

* Lets factorize it using the formula

∵ a = 1 , b = 2 , c = -1

∵ x = [-b ± √(b² - 4ac)]/2(a) ⇒ formula of quadratic equation

∵ b² - 4ac = 2² - 4(1)(-1) = 4 - -4 = 4 + 4 = 8

∵ √8 = 2√2

∴ cos(x) = [-2 ± 2√2]/2(1) = [-2 ± 2√2]/2 ⇒ ÷ 2 up and down

∴ cos(x) = -1 ± √2

* cos(x) = -1 + √2 ⇒ positive value and cos(x) = -1 - √2 ⇒ negative value

∵ x lies on 1st or 4th quadrant

∴ cos(x) must be positive according to the ASTC rule

∴ We will rejected the negative value

* Now lets find the values of angle x

∵ cos(x) = -1 + √2

∴ x = cos^-1(-1 + √2) = 1.1437 ≅ 1.144 ⇒ approximated to the nearest

3 decimal place

* The angle in the 1st quadrant is 1.144 and in the

4th quadrant is-1.144

∴ The answer is (a)

4 0

3 years ago

How do you know if a rate of change is positive or negative?

Paha777 [63]

If the quantity u have decreased with time so the rate is negative and vise versa

8 0

3 years ago

At the market, Ms. Winn bought 3/4 lb of grapes and 5/8 lb of cherries.

wolverine [178]

Answer:

Part a) Ms. Winn bought 12 oz of grapes

Part b) Ms. Winn bought 10 oz of cherries

Part c) Ms. Winn bought 2 oz more of grapes than cherries

Part d) Mr. Phillips bought 6 more ounces of fruit than Ms. Winn

Step-by-step explanation:

Part a) How many ounces of grapes did Ms. Winn buy?

Remember that

$1\ lb=16\ oz$

To convert lb to oz multiply by 16

we know that

Ms. Winn bought 3/4 lb of grapes

Convert lb to oz

Multiply by 16

$\frac{3}{4}\ lb= \frac{3}{4}(16)=12\ oz$

therefore

Ms. Winn bought 12 oz of grapes

Part b) How many ounces of cherries did Ms. Winn buy?

Remember that

$1\ lb=16\ oz$

To convert lb to oz multiply by 16

we know that

Ms. Winn bought 5/8 lb of cherries

Convert lb to oz

Multiply by 16

$\frac{5}{8}\ lb= \frac{5}{8}(16)=10\ oz$

therefore

Ms. Winn bought 10 oz of cherries

Part c) How many more ounces of grapes than cherries did Ms. Winn buy?

Subtract the ounces of cherries from the ounces of grapes

so

$12\ oz-10\ oz=2\ oz$

therefore

Ms. Winn bought 2 oz more of grapes than cherries

Part d) If Mr. Phillips bought 1 3/4 pounds of raspberries, who bought more fruit, Ms. Winn or Mr. Phillips? How many ounces more?

Remember that

$1\ lb=16\ oz$

To convert lb to oz multiply by 16

we know that

Mr. Phillips bought 1 3/4 pounds of raspberries

Convert mixed number to an improper fraction

$1\frac{3}{4}\ lb=\frac{1*4+3}{4}=\frac{7}{4}\ lb$

Convert lb to oz

Multiply by 16

$\frac{7}{4}\ lb= \frac{7}{4}(16)=28\ oz$

Remember that

Ms. Winn bought 12 oz of grapes and 10 oz of cherries

In total Ms. Winn bought

12 oz+10 oz=22 oz -----> total ounces of fruit

so

Mr. Phillips bought 28 oz of fruit

Ms. Winn bought 22 oz of fruit

$28\ oz > 22\ oz$

Find the difference

$(28\ oz-22\ oz)=6\ oz$

therefore

Mr. Phillips bought 6 more ounces of fruit than Ms. Winn

4 0

3 years ago

Possible answers: A) 336cm B) 110m C) 588m D) 1,029m

kozerog [31]

Answer: The area of the enclosure is 1,029 square meters

Step-by-step explanation: The first thing is to use the unit of measurement required by the question, and to do this we need to convert what we have to what is required.

If the scale of the diagram is given as every 4cm represents 7m, that means, every unit of the actual measurement would be given as

(X/4) x 7

For the length of the enclosure, we can determine that as follows;

Length = (28/4) x 7

Length = 7 x 7

Length = 49m

And for the width,

Width = (12/4) x 7

Width = 3 x 7

Width = 21m

Therefore, the area is calculated as,

Area = L x W

Area = 49 x 21

Area = 1029 m²

5 0

2 years ago

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Slope intercept form 8x-6y=18￼

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