Suppose X and Y are independent N(0; 1) random variables.
(a) Find P(X^2 < 1).
(b) Find P(X^2 + Y^2 < 1)
Answers
The required probability is obtained as:
(a) P(X^2 < 1) = 0.8413.
(b) P(X^2 + Y^2 < 1) = 0.7854.
(a) We know that X^2 follows a chi-squared distribution with 1 degree of freedom, which can be written as X^2 ~ chi-squared(1). Therefore, we can find P(X^2 < 1) as:
P(X^2 < 1) = P(Z < √1) (where Z ~ chi-squared(1))
Since the square of a standard normal distribution is a chi-squared distribution with 1 degree of freedom, we can rewrite the above equation as:
P(X^2 < 1) = P(Z < 1) (where Z ~ N(0,1))
Using a standard normal distribution table or calculator, we can find that P(Z < 1) = 0.8413. Therefore, P(X^2 < 1) = 0.8413.
(b) We can rewrite X^2 + Y^2 < 1 as the inequality r^2 < 1, where r is the distance from the origin to the point (X,Y) in the xy-plane. Therefore, we need to find the probability that the point (X,Y) falls within the unit circle centered at the origin.
We can use polar coordinates to express X and Y as:
X = Rcosθ
Y = Rsinθ
where R is the distance from the origin to (X,Y), and θ is the angle between the positive x-axis and the line connecting the origin and (X,Y). Since X and Y are independent N(0,1) random variables, R^2 = X^2 + Y^2 follows a chi-squared distribution with 2 degrees of freedom, which can be written as R^2 ~ chi-squared(2).
Therefore, we can find P(X^2 + Y^2 < 1) as:
P(X^2 + Y^2 < 1) = P(R^2 < 1) (where R^2 ~ chi-squared(2))
Using the cumulative distribution function (CDF) of the chi-squared distribution with 2 degrees of freedom, we can find that:
P(R^2 < 1) = 0.7854
Therefore, P(X^2 + Y^2 < 1) = 0.7854.
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Related Questions
whats the pattern 729, -243, 81, -27, 9
Answers
Answer:
This is the product of three nines (three factors of 9). 9 × 9 × 9 × 9 = 6561 --> This is the product of four nines (four factors of 9). It's a common mistake among educators to state that 81 is what we get by "multiplying nine by itself twice".
999% is 2345 of what number?
Willing to give 5 stars, a thanks, and a brainliest (probably)
Answers
Answer:
4.26
Step-by-step explanation:
Answer:
4.26
Step-by-step explanation:
True False: ANOVA, Part II. Determine if the following statements are true or false, and explain your reasoning for statements you identify as false. If the null hypothesis that the means of four groups are all the same is rejected using ANOVA at a 5% significance level, then (a) we can then conclude that all the means are different from one another. (b) the standardized variability between groups is higher than the standardized variability within groups. (c) the pairwise analysis wi identify at least one pair of means that are significantly different. (d) the appropriate o to be used in pairwise comparisons is 0.05 /4 0.0125 since there are four groups.
Answers
(a) False. If the null hypothesis is rejected using ANOVA, it means that there is significant evidence to suggest that at least one of the means is different from the others.
However, it does not necessarily mean that all the means are different from one another. Additional tests such as pairwise comparisons are needed to determine which specific means are significantly different from one another.
(b) True. When ANOVA is used, the F-statistic is calculated by dividing the variability between groups by the variability within groups. Therefore, if the null hypothesis is rejected, it means that the variability between groups is higher than the variability within groups.
(c) True. Pairwise comparisons are used to determine which specific means are significantly different from one another. If the null hypothesis is rejected using ANOVA, then pairwise comparisons will identify at least one pair of means that are significantly different.
(d) True. When conducting pairwise comparisons, it is important to adjust the significance level to account for the multiple comparisons being made. In this case, there are four groups, so the appropriate level of significance is 0.05/4 = 0.0125. This helps to control the overall type I error rate.
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A water tank at Camp Newton holds 1200 gallons of water at time t = 0. During the time interval Osts 18 hours, water is pumped into the tank at the rate
W(t) = 95Vt sin^2 (t/6) gallons per hour During the same time interval water is removed from the tank at the rate R(t) = 275 sin^2 (1/3) gallons per hour a. Is the amount of water in the tank increasing at time t = 15? Why or why not?
b. To the nearest whole number, how many gallons of water are in the tank at time t = 18? c. At what time t, for 0 st 18, is the amount of water in the tank at an absolute minimum? Show the work that leads to your conclusion d. For t > 18, no water is pumped into the tank, but water continues to be removed at the rate R(C) until the tank becomes empty. Write, but do not solve, an equation involving an integral expression that can be used to find the value of k.
Answers
(a)The amount of water in the tank is increasing.
(b)Evaluate \(\int\limits^{18}_0(W(t) - R(t)) dt\) to get the number of gallons of water in the tank at t = 18.
(c)Solve part (b) to get the absolute minimum from the critical points.
(d)The equation can be set up as \(\int\limits^k_{18}-R(t) dt = 1200\) and solve this equation to find the value of k.
What is the absolute value of a number?
The absolute value of a number is its distance from zero on the number line. It represents the magnitude or size of a real number without considering its sign.
To solve the given problems, we need to integrate the given rates of water flow to determine the amount of water in the tank at various times. Let's go through each part step by step:
a)To determine if the amount of water in the tank is increasing at time t = 15, we need to compare the rate of water being pumped in with the rate of water being removed.
At t = 15, the rate of water being pumped in is given by \(W(t) = 95Vt sin^2(\frac{t}{6})\) gallons per hour. The rate of water being removed is \(R(t) = 275 sin^2(\frac{1}{3})\) gallons per hour.
Evaluate both rates at t = 15 and compare them. If the rate of water being pumped in is greater than the rate of water being removed, then the amount of water in the tank is increasing. Otherwise, it is decreasing.
b) To find the number of gallons of water in the tank at time t = 18, we need to integrate the net rate of water flow from t = 0 to t = 18. The net rate of water flow is given by the difference between the rate of water being pumped in and the rate of water being removed. So the integral to find the total amount of water in the tank at t = 18 is:
\(\int\limits^{18}_0(W(t) - R(t)) dt\)
Evaluate this integral to get the number of gallons of water in the tank at t = 18.
c)To find the time t when the amount of water in the tank is at an absolute minimum, we need to find the minimum of the function that represents the total amount of water in the tank. The total amount of water in the tank is obtained by integrating the net rate of water flow over the interval [0, 18] as mentioned in part b. Find the critical points and determine the absolute minimum from those points.
d. For t > 18, no water is pumped into the tank, but water continues to be removed at the rate R(t) until the tank becomes empty. To find the value of k, we need to set up an equation involving an integral expression that represents the remaining water in the tank after time t = 18. This equation will represent the condition for the tank to become empty.
The equation can be set up as:
\(\int\limits^k_{18}-R(t) dt = 1200\)
Here, k represents the time at which the tank becomes empty, and the integral represents the cumulative removal of water from t = 18 to t = k. Solve this equation to find the value of k.
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Suppose you are learning about infants for a class and need to write a paper on one baby. From a list of births over the past year at a hospital, choose one baby at random. Which of the following events are mutually exclusive (disjoint)?
A. The child having blue eyes and weighing over 10 pounds.
B. The child having brown hair and being a girl.
C. The child being born on a Wednesday and on the weekend.
D. The child being a boy and being over 20 inches tall.
ANSWER IS C
Answers
Answer:
C
Step-by-step explanation:
Don't listen to those *other* answers, clearly the correct one is c
(kidding kidding xD)
just tell me whether to use sin, cos, or tan for number 14 you don’t have to solve
Answers
SOLUTION
We use sin
This is because the opposite and the adjacent sides are where we have known and unknown values
You can use the code SOH/ CAH / TOA to make the right choice of trigonometric ratio to use
What are the terms of y-y-2+2
Answers
Answer:
2y is your answer i think
Step-by-step explanation:
sadie has already written 1 page, and she expects to write 3 pages for every additional hour spent writing. How many hours will sadie have to spend writing this week in order to have written a total of 49 pages
Answers
Let's call x to the number of hours
She writes 3 pages per hour, then after x hours she writes 3x pages
She already wrote 1 page and wants to write a total of 49, so she has to write 49 - 1 = 48 pages.
Then, the equation is:
3x = 48
x = 48/3
x = 16 hours
Reid is a novice cook trying to perfect his baked potato recipe. He decides to put many potatoes in the oven and invites all his friends over. Every few minutes he takes a potato out of the oven and asks some friends to rate its taste on a scale from 1 to 10, where 10 is excellent. For each taste, Reid writes down the number of minutes the potato spent in the oven, x, as well as its rating from 1 to 10, y. The least squares regression line of this data set is:
y=0. 009x+3. 43
Answers
The least squares regression line predicts a 0.534 increase in score for each minute the potatoes are in the oven.
If the data shows a weak relationship between two variables, the line that best fits this linear relationship is called a least-squares regression line, and it minimizes the vertical distance of the data points from the regression line. The term "least squares" is used because it is the minimum of the sum of squared errors, also known as "variance".
In a regression analysis, the dependent variable is shown on the vertical y-axis and the independent variable is shown on the horizontal x-axis. These specifications form the equation for the line of best fit, determined by least squares.
According to the Question:
Reid notes the x number of minutes the potatoes were in the oven and a y rating from 1 to 10. The least squares regression line for this data set is:
y = 0. 009x+3. 43
For each additional minutes a potatoes spends in the oven, then the least square regression line predicts its rating will increase by 0.534.
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You have a balance of 17,426 on your credit card. Your minimum monthly payment is 461 . If your interest rate is 15.5%, how many years will it take to pay off your card assuming you don't add any debt? Enter your response to two decimal places (ex: 1.23)
Answers
With a credit card balance of $17,426, a minimum monthly payment of $461, and an interest rate of 15.5%, we need to calculate the number of years it will take to pay off the card without adding any additional debt.
To determine the time required to pay off the credit card, we consider the monthly payment and the interest rate. Each month, a portion of the payment goes towards reducing the balance, while the remaining balance accrues interest.
To calculate the time needed for repayment, we track the decreasing balance each month. First, we determine the interest accrued on the remaining balance by multiplying it by the monthly interest rate (15.5% divided by 12).
We continue making monthly payments until the remaining balance reaches zero. By dividing the initial balance by the monthly payment minus the portion allocated to interest, we obtain the number of months needed for repayment. Finally, we divide the result by 12 to convert it into years.
In this scenario, it will take approximately 3.81 years to pay off the credit card (17,426 / (461 - (17,426 * (15.5% / 12))) / 12).
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The Point class represents x,y coordinates in a Cartesian plane. Which line of code appears completes this operator which transforms a Point by dx and dy? (Members written inline for this problem.) class Point { int x_{0}, y_{0};public: Point(int x, int y): x_{x}, y_{y} {} int x() const { return x_; } int y() const { return y_; }};Point operator+(int dx, int dy) { return _________________________;}
Answers
The correct line of code that completes this operator which transforms a Point by dx and dy is shown below: Point operator+(int dx, int dy) { return Point(x_+dx,y_+dy);}Note that the function operator+ takes two arguments: an integer dx and an integer dy.
The function returns a point, which is created by adding dx to x and dy to y.The completed code is shown below:class Point { int x_{0}, y_{0};public: Point(int x, int y): x_{x}, y_{y} {} int x() const { return x_; } int y() const { return y_; }};Point operator+(int dx, int dy) { return Point(x_+dx,y_+dy);}Therefore, the correct answer is: `Point(x_+dx,y_+dy)`
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Help me with this please
Answers
Step-by-step explanation:
as, the traingle is equilateral triangle Which means it's all sides are equal
therefore, VT and BT are x
→ BV = VT = BT = x
hope this answer helps you dear take care!
Answer please correctly !!!!!!!!!!!!! Will mark brainliest !!!!!!!!!!!!!!!!!!!
Answers
Answer:
54°
Step-by-step explanation:
If GJ is an angle bisector that the 2 angles are congruent so their measures are equal
7x -9 = 2x +36
7x -2x = 36 +9
5x = 45
x= 45/5 = 9
measure of angle FGH = 7x-9 = 7*9 -9 = 54°
determine whether each set of numbers can be measures of the sides of a triangle. if so, classify the triangle as acute, obtuse, or right. justify your answer. 10, 11
Answers
The set of numbers {10, 11} can be the measures of the sides of a triangle.
To determine whether each set of numbers can be the measure of the sides of a triangle or not, we need to apply the triangle inequality theorem.
The theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
If we consider 10, 11 as the measures of the sides of a triangle, then the sum of any two sides must be greater than the third side.
The possible cases are 10 + 11 > x, where x is the third side.=> x < 21
This implies that the third side must be less than 21 to form a triangle.
Since there are infinitely many possible values of the third side, it can be any value between 1 and 20 inclusive.
Thus, we can classify the triangle based on the length of the longest side as follows:
If the third side is less than 10, the triangle is acute.
If the third side is equal to 10, the triangle is right-angled.
If the third side is greater than 10 and less than 11, the triangle is obtuse.
Therefore, the set of numbers {10, 11} can be the measures of the sides of a triangle.
The triangle can be classified as obtuse with the given measures of sides.
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alculate the double integral. 5x sin(x y) da, r = 0, 6 × 0, 3 r
Answers
The value of the double integral is -1/6 * sin(18) + 3.
To calculate the double integral of 5x * sin(xy) with respect to da (area element), over the region r defined as 0 ≤ x ≤ 6 and 0 ≤ y ≤ 3, we can set up and evaluate the integral as follows:
∬r 5x * sin(xy) da
The integral is taken over the region r, which is a rectangle with sides of length 6 and 3, respectively.
∬r 5x * sin(xy) da = ∫₀³ ∫₀⁶ 5x * sin(xy) dxdy
To evaluate this integral, we perform the integration with respect to x first, followed by y.
∫₀⁶ 5x * sin(xy) dx = [-cos(xy)]₀⁶ = -cos(6y) + 1
Now, we integrate this result with respect to y:
∫₀³ (-cos(6y) + 1) dy = [-1/6 * sin(6y) + y]₀³ = (-1/6 * sin(18) + 3) - (0 + 0) = -1/6 * sin(18) + 3
Therefore, the value of the double integral ∬r 5x * sin(xy) da, over the region r defined as 0 ≤ x ≤ 6 and 0 ≤ y ≤ 3, is -1/6 * sin(18) + 3.
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for geometry:(
will give brainist
Answers
Y= 39
Why?
(6x+15) = (7x-15) solve and get 21 then
7*21-15= 141 take 141 and subtract 180
180-141= 39
Hope this Helps :D
Answer:
x=21,y=159
Step-by-step explanation:
6x+15=7x-6 ( vertically opposite angles)
x=21°
21°+y = 180° (straight angle)
Someone help me pleaseeee
Answers
For the given right triangle, L = 66°, LK = 3.56, and the hypotenuse, LM = 8.75.
What is right triangle?A right triangle, right-angled triangle, or more formally an orthogonal triangle—previously known as a rectangled triangle—is a triangle with one angle that is a right angle (i.e., a 90-degree angle), meaning that two of its sides are perpendicular. The foundation of trigonometry is the relationship between the sides and other angles of a right triangle.
The hypotenuse is the side that faces the right angle (side c in the figure). Legs are the sides that are next to the right angle (or catheti, singular: cathetus). Side a can be thought of as the side that is adjacent to angle B and opposite to (or opposite) angle A, whereas side b is the side that is adjacent to angle A and opposite to angle B.
Given a right triangle with base = 8 and an angle = 24°
Using angle sum property of triangle
24 + 90 + angle L = 180
L = 66°
Using Sohcahtoa
Tangent: tan(θ) = Opposite / Adjacent
tan(24°) = LK/8
0.445 = LK/8
LH = 8 × 0.445
LK = 3.56
LM = \(\sqrt{MK^2 + LK^2}\)
LM = \(\sqrt{8^2 + 3.56^2}\)
LM = 8.75
Thus, For the given right triangle, L = 66°, LK = 3.56, and the hypotenuse, LM = 8.75.
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i'll give you the brainliest!! it's not hard
Answers
Explanation:
When plotting points you start from the x-axis first, then the y-axis second.
From the x-axis we can see that the point is on -1. And from the y-axis we can see that the point is on 4. Therefore the point is (-1,4)
I’ll give out brainliest to whoever explains this.
Answers
Simplity.
(x² – 2y + 7) (3x – 3)
A. Ty – 177 +55 - 21
B. 73 – 179² +5y-21
C. 77-7y2 +5y-21
D. Ty - 2y +557-21
-
Answers
Step byCombine any like terms on each side of the equation: x-terms with x-terms and constants with constants. Arrange the terms in the same order, usually x-term before constants. If all of the terms in the two expressions are identical, then the two expressions are equivalent.
Construct a confidence interval for p1-p2 at the given level of confidence.
x1=30, n1=235, x2=39, n2=294, 90 % confidence
Answers
Using the z-distribution, the 90% confidence interval for the difference of proportions is given by: (-0.0534, 0.0434).
What is the mean and the standard error for the distribution of differences?For each sample, the mean and the standard error are given as follows:
\(p_1 = \frac{30}{235} = 0.1277, s_1 = \sqrt{\frac{0.1277(0.8723)}{235}} = 0.0218\).\(p_2 = \frac{39}{294} = 0.1327, s_1 = \sqrt{\frac{0.1327(0.8673)}{294}} = 0.0198\).For the distribution of differences, they are given by:
\(\overline{p} = p_1 - p_2 = 0.1277 - 0.1327 = -0.005\).\(s = \sqrt{s_1^2 + s_2^2} = \sqrt{0.0218^2 + 0.0198^2} = 0.0294\)What is the confidence interval?The interval is given by:
\(\overline{p} \pm zs\)
In this problem, we have a 90% confidence level, hence\(\alpha = 0.9\), z is the value of Z that has a p-value of \(\frac{1+0.9}{2} = 0.95\), so the critical value is z = 1.645.
Hence the bounds of the interval are:
\(\overline{p} - zs = -0.005 - 1.645(0.0294) = -0.0534\)
\(\overline{p} + zs = -0.005 + 1.645(0.0294) = 0.0434\)
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How many times do you need to regroup to add 873 +467
Answers
Answer:
0 times
Step-by-step explanation:
What is the value of x in3 √ x 9?
Answers
In the given equation x = 9 is the value that satisfies the equation.
What are Square root?The square root of a number is a value that, when multiplied by itself, equals the original number. For example, the square root of 9 is 3, because \(3*3 = 9\). The symbol for square root is "√" and it can also be represented as "sqrt".
In the given equation \(x = 9\) is the value that satisfies the equation.
To find the value of x in the equation \(3 \sqrt{x} = 9\), we first need to square both sides of the equation to eliminate the square root.
\(3 \sqrt{x} = 9\)
Squaring both sides:
\((3 \sqrt{x} )^2 = 9^2\)
\(x = 9^2 / 3^2 = 9^2 / 9 = 9\)
Therefore, x = 9 is the value that satisfies the equation.
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A plane is making its descent to the airport. The distance from the plane to the front edge of a runway is 500 meters and an angle of depression of 40 degrees
To the
nearest hundredth, what is the altitude of the plane?
Answers
Answer:
im not at this stage yet
Step-by-step explanation:
bye felicia!
if one zero of the following polynomial 8x^2-13x-4k is the reciprocal of the other, then find k
Answers
Answer:
Step-by-step explanation:
hello :
if x1 and x2 this zero for ax²+bx+c you have : x1 = 1 /x2 means :x1 × x2 =1
but : x1 × x2 = c /a
in this exercice : a =8 and b = -13 and c = - 4k
so : c/a =-4k/8
so : -4k/8 = 1
-4k = 8 k = 8/-4 k = -2
this polynomial is : 8x²-13x +8
Please help!!!!!!!!!!!!!!!!!!
Answers
Answer: -15a+20c
multiply by -5
1. In a right triangle, the lengths of the legs are a and b. Find the length of a hypotenuse, if: a=1, b=1; 2. In a right triangle, the length of a hypotenuse is c and the length of one leg is a. Find the length of the other leg, if: c=5, a=3;
Answers
Answer:
1. \(c = \sqrt{2}\).
2. b = 4.
Step-by-step explanation:
To solve these two questions, keep the Pythagorean Theorem in mind: \(a^2 + b^2 = c^2\). Also remember that measurements cannot be negative, so we will disregard the negative answers.
1. a = 1, and b = 1. c = ?
\(1^2 + 1^2 = c^2\)
\(1 + 1 = c^2\\\)
\(c^2 = 1 + 1\\c^2 = 2\\\sqrt{c^2} = \sqrt{2}\\c = \sqrt{2}\)
2. a = 3, c = 5. b = ?
\(3^2 + b^2 = 5^2\\9 + b^2 = 25\\b^2 = 16\\\sqrt{b^2} = \sqrt{16}\\b = 4\)
Hope this helps!
HURRY I NEED THIS IN 20 mins please help me
Write an equation for the function g whose graph is the graph off
f(x) = -100(1.05)
translated to the right 5 units and up 50 units.
Answers
Answer:
f(x) = -100(1.05)^(x+5) + 50
Step-by-step explanation:
Here, we want to write an equation that represents the transformation
5 units right means 5 units on the positive x-axis
50 units up means we add this to the total y value
So the resulting equation will be;
f(x) = -100(1.05)^(x+5) + 50
Please explain how to do this.
Answers
Answer:
\((-3,-5)\)
Step-by-step explanation:
We can see that the coordinates of \(X\) is \((1,-4)\). \(X'\) is just the translation of the coordinates of \(X\) by \((x -4, y -1)\).
\((1 -4, -4 -1) \\ (-3, -5)\)
If you don’t know pease be mature and don’t answer I really need the answers
What is the equation of the line that has a slope of 2 and goes through the point (4, -1)
y = 2x - 9
y = 2x - 5
y = 2x - 1
y = 2x + 9
Write the equation of the graph.
y−3=12(x+1)
y+3=−12(x−1)
y=12x+5
y−3=−12(x+1)
What is the slope of a line whose equation is 3x−4y=8
?
−43
−34
34
43
What is the equation below in slope-intercept form?
15x−3y=−12
y = -4x + 8
y = -15x -12
x = 3y - 12
x - y = 6
Answers
Answer:
1. Given the slope is 2, the equation now looks like y = 2x +b Given P (−1,4) we can solve for the b variable by substituting the x and y cordinates. 4 = 2 ⋅ (− 1) + b 4 = − 2 + b b = 6 Therefore the equation that has the slope of 2 and passes through the point (− 1,4) is y = 2x + 9 so it is d
2. i you can do that.
3. The equation for the slope of a line and the points also called point slope form of equation of a straight line is given by: y − y1 = m (x − x1) Whereas the slope-intercept form the equation of the line is given by: y = mx + b. Where b is the y-intercept.
4.In the formula,y = the y-coordinate of any point on the line,m = slope,x = the x-coordinate of any point on the line,and b = ...
Plug the slope and y-intercept into the formula. Remember,the slope is equal to the rise over the run.
You need to know these