Anystate Auto Insurance Company took a random sample of 370 insurance claims paid out during a 1-year period. The average claim paid was $1570. Assume \(\sigma\ math=$250\). Find 0.90 and 0.99 confidence intervals for the mean claim payment.

Mutually exclusive versus independent. The two-way table summarizes data on the gender and eye color of students in a college statistics class. Imagine choosing a student from the class at random. Define event A: student is male, and event B: student has blue eyes. \(\text{Gender}\ \text{Eye color}\begin{array}{l|c|c|c} & \text { Male } & \text { Female } & \text { Total } \\ \hline \text { Blue } & & & 10 \\ \hline \text { Brown } & & & 40 \\ \hline \text { Total } & 20 & 30 & 50 \end{array}\)

Copy and complete the two-way table so that events A and B are mutually exclusive.

A random variable X has the discrete uniform distribution
\(f(x;k)=\frac{1}{k},x=1,2...,k\)
\(f(x;k)=0,\) elsewhere.
Show that the moment-generating function of X is
\(\displaystyle{M}_{{{x}}}{\left({t}\right)}={\frac{{{e}^{{{t}}}{\left({1}-{e}^{{{k}{t}}}\right)}}}{{{k}{\left({1}-{e}^{{{t}}}\right)}}}}\).

You randomly survey students about participating in the science fair. The two-way table shows the results. How many female students do not participate in the science fair?
\(\begin{array}{|c|c|}\hline & \text{No} & \text{Yes} \\ \hline \text{Gender} \\ \hline \text{Female} & 15 & 22 \\ \hline \text{Male} & 12 & 32 \\ \hline \end{array}\)

In 1912 the Titanic struck an iceberg and sank on its first voyage. Some passengers got off the ship in lifeboats, but many died. The following two-way table gives information about adult passengers who survived and who died, by class of travel.

\(\begin{array} {lc} & \text{Class} \\ \text {Survived} & \begin{array}{c|c|c|c} & \text { First } & \text { Second } & \text { Third } \\ \hline \text { Yes } & 197 & 94 & 151 \\ \hline \text { No } & 122 & 167 & 476 \end{array}\ \end{array}\)

Suppose we randomly select one of the adult passengers who rode on the Titanic. Define event D as getting a person who died and event F as getting a passenger in first class. Find P (not a passenger in first class and survived).

Compute the distribution of \(X+Y\) in the following cases:
X and Y are independent normal random variables with respective parameters \((\mu_{1},\sigma_{1}^{2}) and (\mu_{2},\sigma_{2}^{2})\).

Let X and Y be independent, continuous random variables with the same maginal probability density function, defined as
\(f_{X}(t)=f_{Y}(t)=\begin{cases}\frac{2}{t^{2}},\ t>2\\0,\ otherwise \end{cases}\)
(a)What is the joint probability density function f(x,y)?
(b)Find the probability density of W=XY. Hind: Determine the cdf of Z.

The joint density function of two continuous random variables X and Y is:
\(f(x,y)=f(x,y)=\begin{cases}cx^{2}e^{\frac{-y}{3}} & 1
Draw the integration boundaries and write the integration only for \(P(X+Y\leq 6)\)

In government data, a household consists of all occupants of a dwelling unit, while a family consists of 2 or more persons who live together and are related by blood or marriage. So all families form households, but some households are not families. Here are the distributions of household size and family size in the United States. \(\text{Number of people}\ \begin{array}{|l|c|c|c|c|c|c|c|} \hline & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline \begin{array}{l} \text { Household } \\ \text { probability } \end{array} & 0.25 & 0.32 & 0.17 & 0.15 & 0.07 & 0.03 & 0.01 \\ \hline \begin{array}{l} \text { Family } \\ \text { probability } \end{array} & 0 & 0.42 & 0.23 & 0.21 & 0.09 & 0.03 & 0.02 \\ \hline \end{array}\)

Let \(H =\) the number of people in a randomly selected U.S. household and \(F =\) the number of people in a randomly chosen U.S. family. Find the expected value of each random variable. Explain why this difference makes sense.

Compute the distribution of \(X+Y\) in the following cases:
X and Y are independent Poisson random variables with means respective \(\lambda_{1} and \lambda_{2}\).

Consider two continuous random variables X and Y with joint density function
\(f(x,y)=\begin{cases}x+y\ o \leq x \leq 1, 0 \leq y \leq 1\\0 \ \ \ \ otherwise\end{cases}\)
\(P(X>0.8, Y>0.8)\) is?

Suppose that X and Y are continuous random variables with joint pdf \(f(x,y)=e^{-(x+y)} 0\) and zero otherwise.
Find \(P(X>Y)\)