A point $P$ is randomly selected from the square region with vertices at $(\pm 2, \pm 2)$. What is the probability that $P$ is within one unit of the origin

Answers

Answer 1

If P = (X, Y) is a point in the given square, then X and Y are i.i.d random variables each with distribution

[tex]\displaystyle P(X = x) = \begin{cases}\dfrac14 & \text{if } -2 \le x \le 2 \\ 0 & \text{otherwise}\end{cases}[/tex]

and so the joint density of X and Y is

[tex]\displaystyle P(X = x, Y = y) = \begin{cases}\dfrac1{16} & \text{if }-2 \le x \le 2 \text{ and } -2 \le y \le 2 \\ 0 &\text{otherwise}\end{cases}[/tex]

We want to find P(X² + Y² ≤ 1). Points that satisfy this inequality lie in the set

R = {(x, y) : -1 ≤ x ≤ 1 and -√(1 - x²) ≤ y ≤ √(1 - x²)}

but we can more easily describe the region in polar coordinates by setting

x = r cos(t) and y = r sin(t)

so that the set R is identical to

R' = {(r, t) : 0 ≤ r ≤ 1 and 0 ≤ t ≤ 2π}

Integrate the joint density over R' :

[tex]\displaystyle P(X^2 + Y^2 \le 1) = \iint_R \frac1{16} \, dx \, dy[/tex]

[tex]\displaystyle P(X^2 + Y^2 \le 1) = \iint_{R'} \frac r{16} \, dr \, dt[/tex]

[tex]\displaystyle P(X^2 + Y^2 \le 1) = \int_0^{2\pi} \int_0^1 \frac r{16} \, dr \, dt[/tex]

[tex]\displaystyle P(X^2 + Y^2 \le 1) = \int_0^{2\pi} \frac{1^2 - 0^2}{32} \, dt[/tex]

[tex]\displaystyle P(X^2 + Y^2 \le 1) = \frac1{32} \int_0^{2\pi} dt[/tex]

[tex]\displaystyle P(X^2 + Y^2 \le 1) = \frac{2\pi-0}{32}[/tex]

[tex]\displaystyle P(X^2 + Y^2 \le 1) = \boxed{\frac{\pi}{16}}[/tex]

Related Questions

Diedre’s sister works as a lifeguard. Last summer she earned $14 per hour. She earned a total of $5,180 working
as a lifeguard last summer. How many hours did Diedre’s sister work as a lifeguard last summer?

Answers

Answer:

370 hours.

Step-by-step explanation:

Divide $5,180/14 to get your answer of the hours she worked.

Answer:

370 hours is how many she worked

14×370= 5,180

hope this helps

Sketch the graph a x=2 and y= -1 then find the point a which the two graphs intersect

Answers

Answer: They intersect at (2,-1)

Step-by-step explanation:

Step-by-step explanation:

I hope my answer help you. Please, rate it as brainlest answer if it is right.

can someone help me please, this is so stressful:/

Answers

X is 2 because 10 x 2 equals 20 -3 equals 17

Answer:

x=2

Step-by-step explanation:

To solve for x, you want to isolate it. To do this, move terms that are NOT x and use the opposite operation to get it to the other side.

10x - 3 = 17

(Add 3 to both sides)

10x = 20

Now isolate x (since 10x is really 10 * x, you divide)

10x=20

x=2

This is how you would solve to get the solution.

Hope this helps :D

PLEASE PLEASE HELP ME OUT

Answers

Answer:

It is 77 degrees

Step-by-step explanation:

Since line M and N are parralell, the angles are the same

Answer:

X=77

HOPE THIS HELPS

- Todo ❤️

Step-by-step explanation:

Alternative angle thingies are equal

11. Commercial jets fly between 30,000 ft and 36,000 ft. About how many hours of growing could
pass before the beanstalk might interfere with commercial aircrafts? Explain how you got your
answer

Answers

An exponential function is a function that increases rapidly as the time increases.

The number of hours that passes before the magical beanstalk interferes with commercial flight is approximately 9.384 hours.

Reasons:

The function that gives the height of a beanstalk is presented as follows;

b(t) = [tex]3^t[/tex]

When the height is 30,000 ft., we have;

30,000 = [tex]3^t[/tex]

㏑(30,000) = t·㏑(3)

[tex]\displaystyle t = \mathbf{\frac{ln(30,000)}{ln(3)}} \approx 9.384[/tex]

The time it would take the beanstalk to grow 30,000 ft., t ≈ 9.384 hours

Similarly, the time it would take the beanstalk to grow to 36,000 ft. is found as follows;

[tex]\displaystyle t_{36,000} = \mathbf{\frac{ln(36,000)}{ln(3)}} \approx 9.55[/tex]

The time it would take the beanstalk to grow 36,000 ft., t ≈ 9.55 hours

The number of hours it will take the beanstalk to grow to 30,000 feet at

which it might interfere with commercial aircrafts is 9.384 hours.

Learn more here:

https://brainly.com/question/13917934

Solve for x.
2x-30/X+60
90
50

Ο
70
20

Answers

Answer:

It's 50

Step-by-step explanation:

Hope it helped youuh

Answer:

x = 50

Step-by-step explanation:

2x-30+x+60 = 180

3x+30 = 180

3x = 150

x = 50

is 10/110 an irrational or rational number

Answers

Step-by-step explanation: The keywords are times and the sum, which tells us that the 2 operations are multiplication and addition, let's translate this math expression into an English phrase by looking at the 2 operation signs "×" and "+" in order to find the words to translate the given math expression into an English phrase that looks in the equation form and an English phrase like this:

7×4+3 = 3 more than the product of 7 and 4.

I hope that my full given answer with my full given step-by-step explanation is very helpful to your own math question about translating this given expression to an English phrase, please mark me as Brainliest after her answer was been reported by someone and got deleted by The Brainly Moderator, take care, always be safe, and have a great rest of the day! :D

Sincerely,

Jason Ta,

The Ambitious of The Brainly And The Role of The TDSB And WHCI Student of The High School.

What is the length of each side of this equilateral triangle?
A) 42
B) 30
C) 15
D) 12

Answers

Answer:

C I think hope you help this my answer

Answer:

Step-by-step explanation:

no explanation

7.0g divided by 5.6g/cm cubed

Answers

Answer:

Should be 1.25cm

Step-by-step explanation:

Find the slope of a line parallel to each given line.
a
-7x = 3y + 12

Answers

Answer:

[tex] - \frac{7}{3} [/tex]

Step-by-step explanation:

Y = mX + c

, since ( -7X = 3y + 12)

, then -3y = 7x + 12

, then y=

[tex] - \frac{7}{3} x - 4[/tex]

Answer:

-7/3

Step-by-step explanation:-7x divided by 12 plus -3y would give you -7/3

Is 666 = 37 positive or negative?

Answers

Answer: Positive (but not sure)

Step-by-step explanation: Are there answers to the questions is so i could help more but if correct hope i helped :)

PLEASE HELP GIVING BRAINLIEST! :)

Answers

Answer:

Step-by-step explanation:

3a+b - 4

Let a =3 and b = 5

3(3) +5 -4

Multiply

9+5-4

Add and subtract

14-4

Answer:

Step-by-step explanation:

a = 3b = 5

= 3a + b - 4

= 3(3) + 5 - 4

= (3 . 3) + 5 - 4

= 9 + 5 - 4

= 9 + 1

= 10

13 cm
5 cm
7 cm
12 cm
Find the surface area.

Answers

Answer: 37cm

Step-by-step explanation: 5cm + 13cm + 7cm + 12cm = 37cm

Answer:

= [tex]-\frac{1}{4}[/tex]

The y-intercept is just 0 (because the line meets at the y axis at 0).

So, using [tex]y=mx+b[/tex] (where m = slope and b = y-intercept),

[tex]y=-\frac{1}{4}x+0[/tex]

but the '+0' is unnecessary so we just say [tex]y=-\frac{1}{4}x[/tex]

Using the discriminate. Find the number of real solutions. 2^2+7x-15=0

Answers

Answer:

= 11/7

Step-by-step explanation:

Write a question that has the quotient of -3/8

Answers

-6/16 hope this helps

Determine which of the lines, if any, are parallel or perpendicular.

Line a: 4y - 12x = 20
Line b: 3y = -x + 2
Linec: y = 3x - 1

Answers

Step-by-step explanation:

ma=3,mb=_1/3, mc=3a and c are parallelb aree perpendicular to a and c

solve pls brainliest

Answers

Answer:

Mixed Number: 5 [tex]\frac{7}{10}[/tex]

Improper Fraction:[tex]\frac{57}{10}[/tex]

Step-by-step explanation:

A mixed number is when you express the decimal as a fraction, 0.7 is the same as [tex]\frac{7}{10}[/tex], so 5[tex]\frac{7}{10}[/tex] is your answer.

An improper fraction is a fraction where the numerator is bigger than the denominator, to do this you multipy the whole number 5, which can be written as [tex]\frac{5}{1}[/tex], by [tex]\frac{10}{10}[/tex].

[tex]\frac{5}{1}[/tex]·[tex]\frac{10}{10}[/tex]=[tex]\frac{50}{10}[/tex]

Then add 0.7, or [tex]\frac{7}{10}[/tex].

[tex]\frac{50}{10}[/tex]+[tex]\frac{7}{10}[/tex]=[tex]\frac{57}{10}[/tex]

Would you please mark my answer brainliest if it helped you? :D

The hypotenuse of a certain number is 5 units and one leg is 3 units long. Find the area of the triangle

Answers

Answer:

Area: 6

Step-by-step explanation:

It's a five, four, three triangle. Multiply 3 by 4, then divide the answer by 2 to find the area.

If 4 gallons of milk cost $16.24, how much would 7 gallon of milk cost

Answers

Answer:

$28.42

Step-by-step explanation:

First let's calculate the cost per gallon.

We can do this by simply dividing the cost of 4 gallons by 4

1 gallon of milk cost 16.24/4 = 4.06

We want to find the cost of 7 gallons.

We can easily do this by multiplying the cost of one gallon by 7

7 gallons of milk cost 7 * 4.06 = $28.42

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