The triangle (call it <em>T </em>) has base and height 4, so its area is 1/2*4*4 = 8. Then the joint density function is
![f_{X,Y}(x,y)=\begin{cases}\frac18&\text{for }(x,y)\in T\\0&\text{otherwise}\end{cases}](https://tex.z-dn.net/?f=f_%7BX%2CY%7D%28x%2Cy%29%3D%5Cbegin%7Bcases%7D%5Cfrac18%26%5Ctext%7Bfor%20%7D%28x%2Cy%29%5Cin%20T%5C%5C0%26%5Ctext%7Botherwise%7D%5Cend%7Bcases%7D)
where <em>T</em> is the set
![T=\{(x,y)\mid 0\le x\le4\land0\le y\le4-x\}](https://tex.z-dn.net/?f=T%3D%5C%7B%28x%2Cy%29%5Cmid%200%5Cle%20x%5Cle4%5Cland0%5Cle%20y%5Cle4-x%5C%7D)
(a) I've attached an image of the integration region.
![P(X](https://tex.z-dn.net/?f=P%28X%3C3%2CY%3C3%29%3D%5Cdisplaystyle%5Cint_0%5E1%5Cint_0%5E3f_%7BX%2CY%7D%28x%2Cy%29%5C%2C%5Cmathrm%20dy%5C%2C%5Cmathrm%20dx%2B%5Cint_1%5E3%5Cint_0%5E%7B4-x%7Df_%7BX%2CY%7D%28x%2Cy%29%5C%2C%5Cmathrm%20dy%5C%2C%5Cmathrm%20dx%3D%5Cfrac12)
(b) <em>X</em> and <em>Y</em> are independent if the joint distribution is equal to the product of their marginal distributions.
Get the marginal distributions of one random variable by integrating the joint density over all values of the other variable:
![f_X(x)=\displaystyle\int_{-\infty}^\infty f_{X,Y}(x,y)\,\mathrm dy=\int_0^{4-x}\frac{\mathrm dy}8=\begin{cases}\frac{4-x}8&\text{for }0\le x\le4\\0&\text{otherwise}\end{cases}](https://tex.z-dn.net/?f=f_X%28x%29%3D%5Cdisplaystyle%5Cint_%7B-%5Cinfty%7D%5E%5Cinfty%20f_%7BX%2CY%7D%28x%2Cy%29%5C%2C%5Cmathrm%20dy%3D%5Cint_0%5E%7B4-x%7D%5Cfrac%7B%5Cmathrm%20dy%7D8%3D%5Cbegin%7Bcases%7D%5Cfrac%7B4-x%7D8%26%5Ctext%7Bfor%20%7D0%5Cle%20x%5Cle4%5C%5C0%26%5Ctext%7Botherwise%7D%5Cend%7Bcases%7D)
![f_Y(y)=\displaystyle\int_{-\infty}^\infty f_{X,Y}(x,y)\,\mathrm dx=\int_0^{4-y}\frac{\mathrm dx}8=\begin{cases}\frac{4-y}8&\text{for }0\le y\le4\\0&\text{otherwise}\end{cases}](https://tex.z-dn.net/?f=f_Y%28y%29%3D%5Cdisplaystyle%5Cint_%7B-%5Cinfty%7D%5E%5Cinfty%20f_%7BX%2CY%7D%28x%2Cy%29%5C%2C%5Cmathrm%20dx%3D%5Cint_0%5E%7B4-y%7D%5Cfrac%7B%5Cmathrm%20dx%7D8%3D%5Cbegin%7Bcases%7D%5Cfrac%7B4-y%7D8%26%5Ctext%7Bfor%20%7D0%5Cle%20y%5Cle4%5C%5C0%26%5Ctext%7Botherwise%7D%5Cend%7Bcases%7D)
Clearly,
, so they are not independent.
Answer:
m = -18
Step-by-step explanation:
-34=m-16
+16. +16
-18 = m
10% off means the jeans will sell for 90% of the original price ( 100% - 10% = 90%)
Multiply the original price by 90%
65.00 x 0.90 = 58.50
He will pay $58.50
Answer:
B
Step-by-step explanation:
So we have the formula:
![V=\pi r^2h](https://tex.z-dn.net/?f=V%3D%5Cpi%20r%5E2h)
And we want to solve it for r.
So, let's first divide both sides by π and h. This will cancel out the right side:
![r^2=\frac{V}{\pi h}](https://tex.z-dn.net/?f=r%5E2%3D%5Cfrac%7BV%7D%7B%5Cpi%20h%7D)
Now, take the square root of both sides:
![r=\sqrt{\frac{V}{\pi h}}](https://tex.z-dn.net/?f=r%3D%5Csqrt%7B%5Cfrac%7BV%7D%7B%5Cpi%20h%7D%7D)
And we're done!
Our answer is B.
I hope this helps!
Answer:
A = $70753
Step-by-step explanation:
![A = P(1 + \frac{r}{n})^{nt}](https://tex.z-dn.net/?f=A%20%3D%20P%281%20%2B%20%5Cfrac%7Br%7D%7Bn%7D%29%5E%7Bnt%7D)
P = $41000
r = 6.25% = 0.0625
n = 1
t = 9
![A = 41000(1 + 0.0625)^9 = 70752.91](https://tex.z-dn.net/?f=A%20%3D%2041000%281%20%2B%200.0625%29%5E9%20%3D%2070752.91)
![A = \$70753](https://tex.z-dn.net/?f=A%20%3D%20%5C%2470753)