The expectation of the number of people getting their correct hats in the hat check problem, using linearity of expectations. n persons receive n hats

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.

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.
${\displaystyle p=\pm }$

What are the mean and standard deviation of a probability density function given by $Pr(X=k)=\frac{{14}^k{e}^{-14}}{k!}$ for $k\in \{0,1,2...\mathrm{\infty }\}$?

Random variable X and Y have the joint PDF
$f_{XY}(x,y)=\{\begin{array}{l}c,x\ge 0,y\ge 0,(x^2+y)\le 1,\\ 0,otherwise\end{array}$
Find the marginal PDF ${\displaystyle f_X\left(x\right)}$.

A population of values has a normal distribution with mean 191.4 and standard deviation of 69.7. A random sample of size ${\displaystyle n=153}$ is drawn.
Find the probability that a sample of size ${\displaystyle n=153}$ is randomly selected with a mean between 188 and 206.6. Round your answer to four decimal places.
P=?

If you are calculating the standard deviation of measurements, how do you determine how many significant figures to use?

Let X and Y be jointly continuous random variables wth joint PDF is given by:
$f_{X,Y}(x,y)=2$ where $0\le y\le x\le 1$
Show the marginal PDF of X and Y.