Let a and b be coprime integers, and

Deriving the confidence interval
${\displaystyle P(-{\mathrm{\Phi }}_{\alpha }^{-1}<X<{\mathrm{\Phi }}_{\alpha }^{-1})=1-2\alpha }$
Let ${\displaystyle P\sim \mathcal{N}(0,1)}$
$\begin{array}{rlrlr} & P(-{\mathrm{\Phi }}_{\alpha }^{-1}<X<{\mathrm{\Phi }}_{\alpha }^{-1})& & & \left(1\right)\\ & P(X<{\mathrm{\Phi }}_{\alpha }^{-1})-P(X<-{\mathrm{\Phi }}_{\alpha }^{-1})& & P(A<X<B)=P(X<B)-P(X<A)& \left(2\right)\\ & \mathrm{\Phi }\left({\mathrm{\Phi }}_{\alpha }^{-1}\right))-\mathrm{\Phi }(-{\mathrm{\Phi }}_{\alpha }^{-1}\left)\right)& & P(X<A)=\mathrm{\Phi }\left(A\right)& \left(3\right)\\ & \mathrm{\Phi }\left({\mathrm{\Phi }}_{\alpha }^{-1}\right))-(1-\mathrm{\Phi }\left({\mathrm{\Phi }}_{\alpha }^{-1}\right)\left)\right)& & \mathrm{\Phi }(-x)=1-\mathrm{\Phi }\left(x\right)& \left(4\right)\\ & \alpha -1+\alpha & & f\left(f^{-1}\right)=id& \left(5\right)\\ & -1\times (1-2\alpha )& & & \left(6\right)\end{array}$

A nonhomogeneous second-order linear equation and a complementary function $y_c$ are given. Find a particular solution of the equation.
${\displaystyle 4x^{2}y{ }^{″}-4x{y}^{\prime }+3y=8x^{\frac{3}{4}}}$

Find a vector function that represents the curve of intersection of the two surfaces of the cylinder x^2+y^2=4 and the surface z=xy.

A quadratic function has its vertex at the point (−7,2)${\displaystyle (-7,2)}$. The function passes through the point (8,3)${\displaystyle (8,3)}$. When written in vertex form, the function is f(x)=a(x−h)2+k${\displaystyle f\left(x\right)=a{(x-h)}^2+k}$, where:

Sally has caught covid but doesn’t know it yet. She is testing herself with rapid antigen kits which have an 80% probability of returning a positive result for an infected person. For the purpose of this question you can assume that the results of repeated tests are independent. 

a) If sally uses 3 test kits what is the probability that at least one will return a positive result? 

b) In 3 tests, what is the expected number of positive results?

c) Sally has gotten her hands on more effective tests, these ones have a 90% probability of returning a positive result for an infected person. If she tested herself
twice with the new tests, how many positive results would she expect to see?

Given an sample of data set X which is normal-distributed to $N(\mu =240,\sigma )$, I want to find the ${\displaystyle 95\mathrm{\%}}$ confidence interval of ${\displaystyle max(\mu -200,0)}$. As the ${\displaystyle max(\mu -200,0)}$ cut the normal-distribution curve at the point of 200, the sample of ${\displaystyle max(\mu -200,0)}$ is not more normal distributed. Therefore what is a reasonable ${\displaystyle 95\mathrm{\%}}$ confidence interval?