Step-by-step explanation:
djzbiznJzlzbzjdkzlz?kskdbdid jaozbizz,#
I'm going to assume the joint density function is
a. In order for 
to be a proper probability density function, the integral over its support must be 1.
b. You get the marginal density 
by integrating the joint density over all possible values of 
:
c. We have
d. We have
and by definition of conditional probability,
e. We can find the expectation of 
using the marginal distribution found earlier.
![E[X]=\displaystyle\int_0^1xf_X(x)\,\mathrm dx=\frac67\int_0^1(2x^2+x)\,\mathrm dx=\boxed{\frac57}](https://tex.z-dn.net/?f=E%5BX%5D%3D%5Cdisplaystyle%5Cint_0%5E1xf_X%28x%29%5C%2C%5Cmathrm%20dx%3D%5Cfrac67%5Cint_0%5E1%282x%5E2%2Bx%29%5C%2C%5Cmathrm%20dx%3D%5Cboxed%7B%5Cfrac57%7D)
f. This part is cut off, but if you're supposed to find the expectation of , there are several ways to do so.
- Compute the marginal density of , then directly compute the expected value.
![\implies E[Y]=\displaystyle\int_0^2yf_Y(y)\,\mathrm dy=\frac87](https://tex.z-dn.net/?f=%5Cimplies%20E%5BY%5D%3D%5Cdisplaystyle%5Cint_0%5E2yf_Y%28y%29%5C%2C%5Cmathrm%20dy%3D%5Cfrac87)
- Compute the conditional density of given
, then use the law of total expectation.
The law of total expectation says
![E[Y]=E[E[Y\mid X]]](https://tex.z-dn.net/?f=E%5BY%5D%3DE%5BE%5BY%5Cmid%20X%5D%5D)
We have
![E[Y\mid X=x]=\displaystyle\int_0^2yf_{Y\mid X}(y\mid x)\,\mathrm dy=\frac{6x+4}{6x+3}=1+\frac1{6x+3}](https://tex.z-dn.net/?f=E%5BY%5Cmid%20X%3Dx%5D%3D%5Cdisplaystyle%5Cint_0%5E2yf_%7BY%5Cmid%20X%7D%28y%5Cmid%20x%29%5C%2C%5Cmathrm%20dy%3D%5Cfrac%7B6x%2B4%7D%7B6x%2B3%7D%3D1%2B%5Cfrac1%7B6x%2B3%7D)
![\implies E[Y\mid X]=1+\dfrac1{6X+3}](https://tex.z-dn.net/?f=%5Cimplies%20E%5BY%5Cmid%20X%5D%3D1%2B%5Cdfrac1%7B6X%2B3%7D)
This random variable is undefined only when 
which is outside the support of , so we have
![E[Y]=E\left[1+\dfrac1{6X+3}\right]=\displaystyle\int_0^1\left(1+\frac1{6x+3}\right)f_X(x)\,\mathrm dx=\frac87](https://tex.z-dn.net/?f=E%5BY%5D%3DE%5Cleft%5B1%2B%5Cdfrac1%7B6X%2B3%7D%5Cright%5D%3D%5Cdisplaystyle%5Cint_0%5E1%5Cleft%281%2B%5Cfrac1%7B6x%2B3%7D%5Cright%29f_X%28x%29%5C%2C%5Cmathrm%20dx%3D%5Cfrac87)
Answer:
y = 2x-3
Step-by-step explanation:
We have 2 points so we can find the slope
m = (y2-y1)/(x2-x1)
= (1--3)/(2-0)
= (1+3)/(2-0)
=4/2
= 2
The y intercept (where x=0) is -3
We can use the slope intercept form of the equation
y= mx+b where m is the slope and b is the y intercept
y = 2x-3
4y + 3 > 23 subtract 3 from both sides
4y > 20 divide both sides by 4
y > 5
-2y > -2 divide both sides by -2 (the order changes)
y < 1
(-oo, 1) ∪ (5, oo)
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SO SORRY IF THESE IS WRONG:(( JUST SAY IT TO ME NOT TO BULLY ME
