Suppose that the random variables X and Y have joint p.d.f.f(x,y)=\begin{cases}kx(x-y),0<x<2,-x<y<x\\0,\ \ \ \ elsewhere\end{cases}Evaluate the constant k.

We wish to estimate what percent of adult residents in a certain county are parents. Out of 600 adult residents sampled, 192 had kids. Based on this, plot a $99\mathrm{\%}$ confidence interval for the proportion of adult residents who are parents in a given county.
Express your answer in the form of three inequalities. Give your answers in decimal fractions up to three places ${\displaystyle <p<}$ Express the same answer using a point estimate and a margin of error. Give your answers as decimals, to three places.
${\displaystyle p=\pm }$

A population of values has a normal distribution with ${\displaystyle \mu =133.5}$ and ${\displaystyle \sigma =5.2}$. You intend to draw a random sample of size ${\displaystyle n=230}$.
Find the probability that a single randomly selected value is between 133.6 and 134.1.
${\displaystyle P\left(133.6<X<134.1\right)=}$?
Write your answers as numbers accurate to 4 decimal places.

What are the mean and standard deviation of a binomial probability distribution with n=15 and ${\displaystyle p=\frac{3}{7}}$?

Random variables $X_1,X_2,...,X_n$ are independent and identically distributed. 0 is a parameter of their distribution.
If $X_1,X_2,...,X_n$ are Normally distributed with unknown mean 0 and standard deviation 1, then $\bar{X}\sim N(\frac{0,1}{n})$. Use this result to obtain a pivotal function of X and 0.

Suppose that X and Y are independent rv's with moment generating functions $M_X(t)$ and $M_Y(t)$, respectively. If Z=X+Y, show that $M_Z(t)=M_X(t)M_Y(t)$.

What are the mean and standard deviation of a binomial probability distribution with n=18 and ${\displaystyle p=\frac{2}{32}}$?

For a population with a mean of ${\displaystyle \mu =100}$ and a standard deviation of ${\displaystyle \sigma =20}$,
Find the X values.
${\displaystyle z=-.40}$.