Answer:
7 centimeters from what i can see
Answer:
Cov (X,Y) = 6
Step-by-step explanation:
hello,
Cov(X,Y) = E(XY) - E(X)E(Y)
we must first find E(XY), E(X), and E(Y).
since X is uniformly distributed on the interval (0,12), then E(X) = 6.
next we find the joint density f(x,y) using the formula
![f(x,y) = g(y|x)f_{X}(x)](https://tex.z-dn.net/?f=f%28x%2Cy%29%20%3D%20g%28y%7Cx%29f_%7BX%7D%28x%29)
this is because f is uniformly distributed on the the interval (0,12)
also since the conditional probability density of Y given X=x, is uniformly distributed on the interval [0,x], then
for 0≤y≤x≤12
thus
.
hence,
![E(X,Y)= \int\limits^{12}_{x=0} \int\limits^x_{y=o} xy\frac{1}{12x} \,dy dx](https://tex.z-dn.net/?f=E%28X%2CY%29%3D%20%5Cint%5Climits%5E%7B12%7D_%7Bx%3D0%7D%20%20%5Cint%5Climits%5Ex_%7By%3Do%7D%20xy%5Cfrac%7B1%7D%7B12x%7D%20%20%5C%2Cdy%20dx)
![E(X,Y)=\frac{!}{24} \int\limits^{12}_{x=0} x^2 \, dx = 24](https://tex.z-dn.net/?f=E%28X%2CY%29%3D%5Cfrac%7B%21%7D%7B24%7D%20%5Cint%5Climits%5E%7B12%7D_%7Bx%3D0%7D%20x%5E2%20%5C%2C%20dx%20%20%3D%2024)
also,
![E(Y) = \int\limits^{12}_{x=0} \int\limits^x_{y=0} y\frac{1}{12x} \, dydx](https://tex.z-dn.net/?f=E%28Y%29%20%3D%20%5Cint%5Climits%5E%7B12%7D_%7Bx%3D0%7D%20%20%5Cint%5Climits%5Ex_%7By%3D0%7D%20%20y%5Cfrac%7B1%7D%7B12x%7D%20%20%5C%2C%20dydx)
![E(Y)=\frac{1}{24}\int\limits^{12}_{x=0} {x} \, dx =3](https://tex.z-dn.net/?f=E%28Y%29%3D%5Cfrac%7B1%7D%7B24%7D%5Cint%5Climits%5E%7B12%7D_%7Bx%3D0%7D%20%7Bx%7D%20%5C%2C%20dx%20%20%3D3)
thus Cov(X,Y) = E(XY) - E(X)E(Y)
= 24 - (6)(3)
= 6
Answer:
4√2 + 2√10
Step-by-step explanation:
Finding the perimeter here involves finding three point-to-point distances and then adding them up.
From (2, 2) to (6, -2) we have a change in x of 4 and a change in y of -4. We use the Pythagorean Theorem to determine the distance, which is the hypotenuse of a right triangle:
√( [4]² + [-4]² ) = 4√2. This is the length of the side connecting (2, 2) and (6, -2).
From (6, -2) to (5, 1), the distance is √( [-1]² + 3² ), or √10.
From (5, 1) to (2, 2), the distance is √( [-3]² + 1² ), or √10.
The perimeter is the sum of these three distances and is:
4√2 + √10 + √10, or 4√2 + 2√10
The odds of 1 coin landing tails is 1/2, for 2 coins it would be 1/4, 3 would be 1/8, 4 would be 1/16, etc.
the way you find this is by doing 1/2^n where n is the number of coins you have. if you plug in 20, you get 1/1,048,576