Mathematics, 14.07.2019 00:00 ChaosMind

The joint density function for random variables x, y, and z is f(x, y, z)= cxyz if 0 ≤ x ≤ 1, 0 ≤ y ≤ 2, 0 ≤ z ≤ 2, and f(x, y, z) = 0 otherwise. (a) find the value of the constant c. (b) find p(x ≤ 1, y ≤ 1, z ≤ 1). (c) find p(x + y + z ≤ 1).

Answers: 1

Show answers

Answers

Answer from: Samanthas6365

In order for f(x,y,z) to be a proper density function, we need to have

$\displaystyle\int_{\mathcal S}f(x,y,z)\,\mathrm dx\,\mathrm dy\,\mathrm dz=1$

where $\mathcal S$ is the support of the joint distribution. Here $\mathcal S$ is the box with dimensions 1x2x2. We have

$\displaystyle\int_{\mathcal S}f(x,y,z)\,\mathrm dx\,\mathrm dy\,\mathrm dz=c\int_{x=0}^{x=1}x\,\mathrm dx\int_{y=0}^{y=2}y\,\mathrm dy\int_{z=0}^{z=2}z\,\mathrm dz$

$=\dfrac{2^2c}2=1\implies c=\dfrac12$

Now since we can write $f_{X,Y,Z}(x,y,z)$ as

$\dfrac{xyz}2=\dfrac x{c_x}\cdot\dfrac y{c_y}\cdot\dfrac z{c_z}$

(where c_xc_yc_z=2 ) we can assign each factor to be the PDF for the corresponding random variable, i.e.

$f_{X,Y,Z}(x,y,z)=f_X(x)\cdot f_Y(y)\cdot f_Z(z)$

which means that X,Y,Z are independent of one another. Then

$P(X\le1,Y\le1,Z\le1)=P(X\le1)\cdot P(Y\le1)\cdot P(Z\le1)$

We can derive the marginal distributions in the same way we solve for above:

$\displaystyle\int_{x=0}^{x=1}\frac x{c_x}\,\mathrm dx=1\implies c_x=\frac12$

$\implies f_X(x)=\begin{cases}2x&\text{for }0\le x\le1\\0&\text{otherwise}\end{cases}$

$\displaystyle\int_{y=0}^{y=2}\frac y{c_y}\,\mathrm dz=\int_{z=0}^{z=2}\frac z{c_z}\,\mathrm dz=1\implies c_y=c_z=2$

$\implies f_Y(y)=\begin{cases}\dfrac y2&\text{for }0\le y\le2\\\\0&\text{otherwise}\end{cases}$

with f_Z(z) defined similarly, so that

$P(X\le 1,Y\le1,Z\le1)=1\cdot\dfrac14\cdot\dfrac14=\dfrac18$

For the final probability, we condition on the event that X=x and Y=y :

$P(X+Y+Z\le1)=P(Z\le1-X-Y)$

$=\displaystyle\iint_{(x,y)}P(Z$

$=\displaystyle\int_{x=0}^{x=1}\int_{y=0}^{y=2}F_Z(1-x-y)f_X(x)f_Y(y)\,\mathrm dy\,\mathrm dx$

The CDF of is easy to find:

$F_Z(z)=\begin{cases}0&\text{for }z$

So we have

$P(X+Y+Z\le1)=\displaystyle\int_{x=0}^{x=1}\int_{y=0}^{y=2}\frac{(1-x-y)^2}4\cdot2x\cdot\frac y2\,\mathrm dx\,\mathrm dy=\frac{23}{72}$

Answer from: Quest

$y> \frac{1}{3}x$ and y≤ -x +4 .

step-by-step explanation:

we are given a graph for system of inequalities.

in order to write the system of inequalities for the given graph, we need to find the inequality for each of the line separately.

let us find the slope of dotted line first.

taking two coordinates on the graph.

let us say (0,0) and (3,1).

let us find the slope of the line now.

slope of the dotted line would be = $\frac{1-0}{3-0} = \frac{1}{3}$ .

y-intercept of dotted line is y=0 only.

plugging slope m= $\frac{1}{3}$ and b=0 in slope-intercept form y=mx+b, we get

$y=\frac{1}{3}x$ .

the line is shaded up of that dotted line.

therefore, for shading up we would take greater than inequality symbol > .

and because it's a dotted line we would take only > symbol in place of = sign in above equation of line.

therefore, inequality for dotted line would be

$y> \frac{1}{3}x$ .

let us take two points (3,1) and (0,4) on the solid line.

slope of the solid line would be = $\frac{4-1}{0-3} = \frac{3}{-3}=-1$ .

for the solid line we have y-intercept b= 4.

plugging slope m= -1 and b=4 in slope-intercept form y=mx+b, we get

y = -1x + 4.

the line is shaded down of that solid line.

therefore, for shading down we would take less than inequality symbol < .

and because it's a solid line we would take ≤ symbol in place of = sign in above equation of line.

therefore, inequality for solid line would be

y≤ -x +4 .
Write a system of inequalities to represent the shaded portion of the graph.

Answer from: Quest

y=3n + 1

step-by-step explanation:

this is wrong because it says the second number is one less and i added one instead

Another question on Mathematics

Mathematics, 20.06.2019 18:04

Is the open sentence 5x-2=3x+2 true when x=2? justify your answer.

Answers: 1

Answer

Mathematics, 21.06.2019 15:00

Find the solution of system of equation y= x-4 y=4x+2

Answers: 3

Answer

Mathematics, 21.06.2019 16:00

5,600 x 10^3 = a) 560 b) 5,000,600 c) 5,600,000 d) 56,000

Answers: 2

Answer

Mathematics, 21.06.2019 21:00

Isabel graphed the following system of equations. 2x – y = 6 y = -3x + 4 she came up with the solution (2,-2). what were the 3 steps she did to get that solution? (make sure they are in the correct order)

Answers: 2

Answer

You know the right answer?

The joint density function for random variables x, y, and z is f(x, y, z)= cxyz if 0 ≤ x ≤ 1, 0 ≤ y...

Questions

Mathematics, 16.12.2020 04:30

if you ask your ex how him and his gf doing and he says great should i take that as an assault lol or am i being too sensitive..?...

Mathematics, 16.12.2020 04:30

One glass of lemonade uses two thirds cup of lemonade mix. How many glasses of lemonade can Stephanie make with 3 cups of lemonade mix?...