Answer:
-14-158b
Step-by-step explanation:
Answer:
Option c
because
others aren't even interior angles
Answer:
0.99
Step-by-step explanation:
Answer:
The value of a and b are 8 and 1 respectively
Step-by-step explanation:
![f_{x,y}(x,y)=\left\{\begin{matrix}cxy ,\text{if} 0 \leq x \leq y \leq 1\\ 0 , \text{otherwise}\end{matrix}\right.](https://tex.z-dn.net/?f=f_%7Bx%2Cy%7D%28x%2Cy%29%3D%5Cleft%5C%7B%5Cbegin%7Bmatrix%7Dcxy%20%2C%5Ctext%7Bif%7D%20%200%20%5Cleq%20x%20%5Cleq%20y%20%5Cleq%201%5C%5C%200%20%2C%20%5Ctext%7Botherwise%7D%5Cend%7Bmatrix%7D%5Cright.)
![\int_{0}^{1}\int_{x}^{1}cxy \text{dy dx}=1\\\\c\int_{0}^{1}x\left ( \frac{y^2}{2}\right )_{x}^{1} \text{ dx}=1\\\\\frac{c}{2}\int_{0}^{1}x\left ( 1-y^2\right )\text{dx}=1\\\\\frac{c}{2}\int_{0}^{1}(x-x^3)\text{dx}=1\\\\\frac{c}{2}\left [ \frac{x^2}{2}-\frac{x^5}{4} \right ]_{0}^1=1\\\\\frac{c}{2}\left [ \frac{1}{2}-\frac{1}{4} \right ]=1\\\\\frac{c}{2}\left ( \frac{1}{4} \right )=1\\\\c=8](https://tex.z-dn.net/?f=%5Cint_%7B0%7D%5E%7B1%7D%5Cint_%7Bx%7D%5E%7B1%7Dcxy%20%5Ctext%7Bdy%20dx%7D%3D1%5C%5C%5C%5Cc%5Cint_%7B0%7D%5E%7B1%7Dx%5Cleft%20%28%20%5Cfrac%7By%5E2%7D%7B2%7D%5Cright%20%29_%7Bx%7D%5E%7B1%7D%20%5Ctext%7B%20dx%7D%3D1%5C%5C%5C%5C%5Cfrac%7Bc%7D%7B2%7D%5Cint_%7B0%7D%5E%7B1%7Dx%5Cleft%20%28%201-y%5E2%5Cright%20%29%5Ctext%7Bdx%7D%3D1%5C%5C%5C%5C%5Cfrac%7Bc%7D%7B2%7D%5Cint_%7B0%7D%5E%7B1%7D%28x-x%5E3%29%5Ctext%7Bdx%7D%3D1%5C%5C%5C%5C%5Cfrac%7Bc%7D%7B2%7D%5Cleft%20%5B%20%5Cfrac%7Bx%5E2%7D%7B2%7D-%5Cfrac%7Bx%5E5%7D%7B4%7D%20%5Cright%20%5D_%7B0%7D%5E1%3D1%5C%5C%5C%5C%5Cfrac%7Bc%7D%7B2%7D%5Cleft%20%5B%20%5Cfrac%7B1%7D%7B2%7D-%5Cfrac%7B1%7D%7B4%7D%20%5Cright%20%5D%3D1%5C%5C%5C%5C%5Cfrac%7Bc%7D%7B2%7D%5Cleft%20%28%20%5Cfrac%7B1%7D%7B4%7D%20%5Cright%20%29%3D1%5C%5C%5C%5Cc%3D8)
Conditional pdf of ![f_{x,y}(x|0.5)](https://tex.z-dn.net/?f=f_%7Bx%2Cy%7D%28x%7C0.5%29)
![f_{x,y}(x|0.5)=\frac{f_{xy}(x,y=0.5)}{f_{y}y}](https://tex.z-dn.net/?f=f_%7Bx%2Cy%7D%28x%7C0.5%29%3D%5Cfrac%7Bf_%7Bxy%7D%28x%2Cy%3D0.5%29%7D%7Bf_%7By%7Dy%7D)
Normalizing pdf of 1
![f_{y}y=\int_{0}^{y} 8yx \text{dx}](https://tex.z-dn.net/?f=f_%7By%7Dy%3D%5Cint_%7B0%7D%5E%7By%7D%208yx%20%5Ctext%7Bdx%7D)
![f_{y}y=8y[\frac{x^2}{2}]_{0}^{y}\\f_{y}y=4y(y^2)\\f_{y}y=4y^3\\f_{x|y}(x|0.5)=\frac{8x(0.5)}{4(0.5)^3}=\frac{2x}{0.25}=8x](https://tex.z-dn.net/?f=f_%7By%7Dy%3D8y%5B%5Cfrac%7Bx%5E2%7D%7B2%7D%5D_%7B0%7D%5E%7By%7D%5C%5Cf_%7By%7Dy%3D4y%28y%5E2%29%5C%5Cf_%7By%7Dy%3D4y%5E3%5C%5Cf_%7Bx%7Cy%7D%28x%7C0.5%29%3D%5Cfrac%7B8x%280.5%29%7D%7B4%280.5%29%5E3%7D%3D%5Cfrac%7B2x%7D%7B0.25%7D%3D8x)
We are given that PDF
is of the form ![ax^b](https://tex.z-dn.net/?f=ax%5Eb)
So, on comparing 8x with ![ax^b](https://tex.z-dn.net/?f=ax%5Eb)
So, a = 8 , b = 1
So, the value of a and b are 8 and 1 respectively