Answer: 1,680 pints
Step-by-step explanation: 210 gallons= 1,680 pints
There is one cake
It is divided among x family members, so each gets:
1/x
The father took 1/5 which is 4 times greater than each one's piece. So:
1/5 = 4(1/x)
1/5 = 4/x
x = 20
There are 20 family members so the answer is none of these
a. 
is a joint density function if its integral over the given support is 1:
so the answer is yes.
b. We should first find the density of the marginal distribution, 
:
Then
or about 0.2019.
For the other probability, we can use the joint PDF directly:
which is about 0.7326.
c. We already know the PDF for 
, so we just integrate:
![E[Y]=\displaystyle\int_{-\infty}^\infty y\,f_Y(y)\,\mathrm dy=\frac15\int_0^\infty ye^{-y/5}\,\mathrm dy=\boxed5](https://tex.z-dn.net/?f=E%5BY%5D%3D%5Cdisplaystyle%5Cint_%7B-%5Cinfty%7D%5E%5Cinfty%20y%5C%2Cf_Y%28y%29%5C%2C%5Cmathrm%20dy%3D%5Cfrac15%5Cint_0%5E%5Cinfty%20ye%5E%7B-y%2F5%7D%5C%2C%5Cmathrm%20dy%3D%5Cboxed5)
R + R - 6 + 4R = 126
6R - 6 = 126
6R = 126 + 6
6R = 132
R = 132/6
R = 22 (hours Ruby worked)
I = R - 6
I = 22 - 6
I = 16 (hours Isaac worked)
S = 4R
S = 4(22)
S = 88 (hours Svetlana worked)
Finally, let's check our three answers to see if their sum is 126 (the given total of hours worked):
22 + 16 + 88 = 38 + 88 = 126
