The area A of a triangle is given

Statistics and Probability

The joint probability distribution of two random variables X and Y is given in the following table.

1. Calculate the marginal probability distribution of X.
2. Calculate the marginal probability distribution of Y.
3. Calculate E( X ) and s.d.( X ).
4. Calculate E( Y ) and s.d.( Y )
5. Calculate cov(X, Y) and corr(X, Y).
6. Calculate the conditional probability distribution of X given Y = -1.
7. Calculate the conditional probability distribution of X given Y = 2.
8. Calculate E( X | Y = -1 ) and var( X | Y = -1 ).
9. Calculate E( X | Y = 2 ) and var( X | Y = 2 ).
10. Verify the Law of Iterated Expectations E(X)=E[E(X|Y)] by using your previous results.
11. Are X and Y independent random variables? Justify your answer based on answers to the preceding questions.

How to graph ${\displaystyle f\left(x\right)=\frac{1-\cos (2x-2)}{{(x-1)}^2}}$ analytically?
I only have an issue when it comes to determining the hole at $x=1$ analytically. Graphing the function on a graphing utility reveals a hole at $x=1$ rather than a vertical asymptote. I know that by simply plugging in the value $1$ for $x$, one will see that both the numerator and denominator equate to zero. This shows that there is at least one factor of $(x-1)$ that can be extracted from the numerator.
I would like to know how to simplify this function analytically in such a way that two factors of $(x-1)$ can be obtained from the numerator. I believe one must make use of an inverse function down the line in order to extract the obvious $(x-1)$ hiding in the ${\displaystyle \cos (2x-2)=\cos \left(2(x-1)\right)}$ term. That is as far as I got in the problem.

What is the instantaneous rate of change of $f\left(x\right)=e^x$ when $x=0$?

y"+3y'+2y=2 if 0<t<5 y(0)=0 and y'(0)=0

What is the formula for finding the length of the hypotenuse of an isosceles triangle, for ANY ANGLE, (or angle < 180), with two unit sides AB = AC = 1 (unit), where BC is the hypotenuse.

Thanks

there are 6 girls and 8 boys at a birthday party. In how many ways can they form six dancing boy-girl couples?