Mathematics, 20.12.2021 09:50, mettababeeeee

A random variables X and Y are distributed according to the joint PDF. The value of constant a = .f(x, y)=ax if 1<=1x<=y<=2 and 0 otherwise

Answers: 3

Show answers

Answers

answered: maddieberridgeowud2s

3/2

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

Explanation:

PDF = probability density function

The given joint PDF is

$f(x,y) = \begin{cases}ax \ \ \ \text{ if } 1 \le x \le y \le 2\\0 \ \ \ \ \ \text{ otherwise}\end{cases}$

Let's focus on the $1 \le x \le y \le 2$ . Specifically the x term for now. Erasing out the y term, we have the inequality $1 \le x \le 2$ which says x is between 1 and 2, inclusive. We have almost the same story for y, but there's another condition attached to it: y must also be equal to or larger than x.

So let's say x = 1.5. This would mean $1.5 \le y \le 2$ . As another example, x = 1.7 leads to $1.7 \le y \le 2$ . In general, we would say $x \le y \le 2$ where x is between 1 and 2.

As x gets bigger, the range of possible y values gets smaller. If x = 2, then y has no choice but to be 2 as well.

-----------------

Based on that, we'll have a double integral that looks like this:

$\displaystyle V = \int_{1}^{2}\int_{x}^{2}f(x,y)dydx\\\\$

The outer integral handles the x terms that range from 1 to 2, describing $1 \le x \le 2$ . Note the dx on the outside. The order of the dy and dx matters.

On the inside, we have the integral for dy ranging from x to 2 to describe the interval $x \le y \le 2$

To have f(x,y) be a PDF, the volume under the f(x,y) surface must be 1, where the volume is based on the bounds set up. So we must have V = 1. We'll use this later.

-----------------

Let's simplify the double integral.

We'll start by computing the inner integral with respect to y.

$\displaystyle V = \int_{1}^{2}\int_{x}^{2}f(x,y)dydx\\\\\displaystyle V = \int_{1}^{2}\int_{x}^{2}\left(ax\right)dydx\\\\\displaystyle V = \int_{1}^{2}\left(axy\Bigg|_{x}^{2}\right)dx\\\\\displaystyle V = \int_{1}^{2}\left(ax(2) - ax(x)\right)dx\\\\\displaystyle V = \int_{1}^{2}\left(2ax - ax^2\right)dx\\\\$

Then we'll finish it off by integrating with respect to x.

$\displaystyle V = \int_{1}^{2}\left(2ax - ax^2\right)dx\\\\\displaystyle V = \left(ax^2 - \frac{1}{3}ax^3\right)\Bigg|_{1}^{2}\\\\\displaystyle V = \left(a(2)^2 - \frac{1}{3}a(2)^3\right) - \left(a(1)^2 - \frac{1}{3}a(1)^3\right)\\\\\displaystyle V = \left(4a - \frac{8}{3}a\right)-\left(a - \frac{1}{3}a\right)\\\\$

$\displaystyle V = 4a - \frac{8}{3}a-a + \frac{1}{3}a\\\\\displaystyle V = 3a - \frac{8}{3}a + \frac{1}{3}a\\\\\displaystyle V = \frac{9}{3}a - \frac{8}{3}a + \frac{1}{3}a\\\\\displaystyle V = \frac{9-8+1}{3}a\\\\\displaystyle V = \frac{2}{3}a\\\\$

Side note: We don't have to worry about the "plus C" integration constant when working with definite integrals.

Recall that V = 1. So,

$\displaystyle V = \frac{2}{3}a\\\\\displaystyle \frac{2}{3}a = 1\\\\\displaystyle a = \frac{3}{2} = 1.5\\\\$

a = 3/2 is the final answer.

answered: Guest

x = -2

y = 848

step-by-step explanation:

plug in the numbers

answered: Guest

we have the parent function given by,

1. the function is translated 5 units to the left.

so, we get, becomes

2. the function is translated 5 units to the right.

so, we get, becomes

3. the function is translated 5 units upwards.

so, we get, becomes

4. the function is translated 5 units downwards.

so, we get, becomes

5. the function is reflected across x-axis.

so, we get, becomes

6. the function is reflected across y-axis.

so, we get, becomes .

step-by-step explanation:

mark brainliest.

Other questions on the subject: Mathematics

Mathematics, 21.06.2019 17:00, laylay7383

The table below shows the height of a ball x seconds after being kicked. what values, rounded to the nearest whole number, complete the quadratic regression equation that models the data? f(x) = x2 + x + 0based on the regression equation and rounded to the nearest whole number, what is the estimated height after 0.25 seconds? feet

Answers: 2

continue

Mathematics, 21.06.2019 18:10, ltuthill

Jordan has $5.37, which he is using to buy ingredients to make salsa. he is buying one red pepper for $1.29 and three pounds of tomatoes. if jordan has exactly the right amount of money he needs, what is the price per pound of the tomatoes? choose the correct equation to represent this real-world problem. solve the equation and verify the reasonableness of your answer. a pound of tomatoes costs .

Answers: 1

continue

Mathematics, 21.06.2019 19:00, naomicervero

Four individuals pool money together to start a new business and agree to split the profits equally. n invests $6,000, x invests $2,000, y invests $8,000 and z invests $4,000. if the profits for the first month were $100,000, y receives than if the profits were divided in proportion to how much they invested.

Answers: 3

continue

Mathematics, 21.06.2019 21:30, fheight01

Name and describe the three most important measures of central tendency. choose the correct answer below. a. the mean, sample size, and mode are the most important measures of central tendency. the mean of a data set is the sum of the observations divided by the middle value in its ordered list. the sample size of a data set is the number of observations. the mode of a data set is its highest value in its ordered list. b. the sample size, median, and mode are the most important measures of central tendency. the sample size of a data set is the difference between the highest value and lowest value in its ordered list. the median of a data set is its most frequently occurring value. the mode of a data set is sum of the observations divided by the number of observations. c. the mean, median, and mode are the most important measures of central tendency. the mean of a data set is the product of the observations divided by the number of observations. the median of a data set is the lowest value in its ordered list. the mode of a data set is its least frequently occurring value. d. the mean, median, and mode are the most important measures of central tendency. the mean of a data set is its arithmetic average. the median of a data set is the middle value in its ordered list. the mode of a data set is its most frequently occurring value.

Answers: 3

continue

Do you know the correct answer?

A random variables X and Y are distributed according to the joint PDF. The value of constant a = .f(...