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3 years ago

10

Suppose you roll n ≥ 1 fair dice. Let X be the random variable for the sum of their values, and let Y be the random variable for

the number of times an odd number comes up. Prove or disprove: X and Y are independent.

1 answer:

noname [10]3 years ago

3 0

<u>Answer:</u>

X and Y are stochastically dependent RVs .

<u>Step-by-step explanation:</u>

Let ,

X = sum of the values that come up after throwing n (≥ 1) fare dice.

Y = number of times an odd number come up.

Let, n = 3

then, P(X =6) = p (say) clearly 0 < p < 1

and P (Y = 3) = $\frac{1}{8}$

And,

P( X = 6, Y = 3) = 0 ≠ $P(X = 6) \times P(Y= 3)$

Hence, X and Y are stochastically dependent RVs

You might be interested in

5 37/40 as a decimal

Luden [163]

Well 5=5
and 37/40
hmm, try to get that 40 into an nice number that is multiplule of 10
if we divide top and bottom by 4 that would work
$( \frac{37}{40})( \frac{ \frac{1}{4} }{ \frac{1}{4} })= \frac{9.25}{10}$
9.25/10=92.5=100=925/1000=925 thousandths=0.925

so the decimal would be 5.925

5 0

4 years ago

Read 2 more answers

Verify that the function u(x, y, z) = log x^2 + y^2 is a solution of the two dimensional Laplace equation u_xx + u_yy = 0 everyw

Daniel [21]

Answer:

The function $u(x,y,z)=log ( x^{2} +y^{2})$ is indeed a solution of the two dimensional Laplace equation $u_{xx} +u_{yy} =0$ .

The wave equation $u_{tt} =u_{xx}$ is satisfied by the function $u(x,t)=cos(4x)cos(4t)$ but not by the function $u(x,t)=f(x-t)+f(x+1)$ .

Step-by-step explanation:

To verify that the function $u(x,y,z)=log ( x^{2} +y^{2})$ is a solution of the 2D Laplace equation we calculate the second partial derivative with respect to x and then with respect to t.

$u_{xx}=\frac{2}{ln(10)}((x^{2} +y^{2})^{-1} -2x^{2} (x^{2} +y^{2})^{-2})$

$u_{yy}=\frac{2}{ln(10)}((x^{2} +y^{2})^{-1} -2y^{2} (x^{2} +y^{2})^{-2})$

then we introduce it in the equation $u_{xx} +u_{yy} =0$

we get that $\frac{2}{ln(10)} (\frac{2}{(x^{2}+y^{2}) } - \frac{2}{(x^{2}+y^{2} ) } )=0$

To see if the functions 1) $u(x,t)=cos(4x)cos(4t)$ and 2) $u(x,t)=f(x-t)+f(x+1)$ solve the wave equation we have to calculate the second partial derivative with respect to x and the with respect to t for each function. Then we see if they are equal.

1) $u_{xx}=-16 cos (4x) cos (4t)$

$u_{tt}=-16cos(4x)cos(4t)$

we see for the above expressions that $u_{tt} =u_{xx}$

2) with this function we will have to use the chain rule

If we call $s=x-t$ and $w=x+1$ then we have that

$u(x,t)=f(x-t)+f(x+1)=f(s)+f(w)$

So $\frac{\partial u}{\partial x}=\frac{df}{ds}\frac{\partial s}{\partial x} +\frac{df}{dw} \frac{\partial w}{\partial x}$

because we have $\frac{\partial s}{\partial x} =1$ and $\frac{\partial w}{\partial x} =1$

then $\frac{\partial u}{\partial x} =f'(s)+f'(w)$

⇒ $\frac{\partial^{2} u }{\partial x^{2} } =\frac{\partial}{\partial x} (f'(s))+ \frac{\partial}{\partial x} (f'(w))$

⇒ $\frac{\partial^{2} u }{ \partial x^{2} } =\frac{d}{ds} (f'(s))\frac{\partial s}{\partial x} +\frac{d}{ds} (f'(w))\frac{\partial w}{\partial x}$

⇒ $\frac{\partial^{2} u }{ \partial x^{2} } =f''(s)+f''(w)$

Regarding the derivatives with respect to time

$\frac{\partial u}{\partial t}=\frac{df}{ds} \frac{\partial s}{\partial t}+\frac{df}{dw} \frac{\partial w}{\partial t}=-\frac{df}{ds} =-f'(s)$

then $\frac{\partial^{2} u }{\partial t^{2} } =\frac{\partial}{\partial t} (-f'(s))=-\frac{d}{ds} (f'(s))\frac{\partial s}{\partial t} =f''(s)$

we see that $\frac{\partial^{2} u }{ \partial x^{2} } =f''(s)+f''(w) \neq f''(s)=\frac{\partial^{2} u }{\partial t^{2} }$

$u(x,t)=f(x-t)+f(x+1)$ doesn´t satisfy the wave equation.

4 0

3 years ago

HELP HELP HELP PLEASEE!!

kotykmax [81]

Answer:

Movie: $4.25

Video Game: $5.25

Step-by-step explanation:

For this problem, we can use system of equations to solve for the rental costs of the movies and video games. Let's use x for movies and y for video games.

Equation 1

5x+3y=37

Equation 2

2x+6y=40

To start, we can use elimination to solve for x and y. Let's cancel out y. To do so, we have to multiply the first equation by 2 to get the y equal.

10x+6y=74

Now that the y are equal, we can subtract both equations.

8x=34

x=4.25

The movie rental cost is $4.25.

Now that we know x, we can plug in to find y.

2(4.25)+6y=40

8.5+6y=40

6y=31.5

y=5.25

The rental cost for video games is $5.25.

4 0

3 years ago

What is f−1(f(x)) = f(f−1(x)) = equal to????? please help

wel

the assumption here is that the function f(x), has an inverse function of f⁻¹(x), so let's assume that is indeed the case, f⁻¹(x) is the inverse function of f(x).

for functions and their inverse, if they're indeed inverse of each other, then

f⁻¹( f(x) ) = x

f( f⁻¹(x) ) = x.

6 0

3 years ago

Olivia ran 6 miles in 51 min. how long does it take her to run 1 mile

Lady bird [3.3K]

Olivia ran 6 miles in 51 minutes. You are required to find the time, in minutes, it takes her to run in 1 mile. The equation to be used in this problem is speed equals distance over time. The solution is as follows:

S = distance/time
S = 6 miles/ 51 minutes
S = 0.118 miles/minutes

To find the time Olivia run in 1 mile, use the value of speed above and equation.
S = distance/time
0.118 miles/minute = 1 mile / time
Time = 1 mile/ 0.118 miles/minute
<u>Time = 8.5 miles</u>

4 0

3 years ago