Explore Random Variables: Examples & Equations

Let X be a random variable with probability density function.
$f(x)=\{\begin{array}{l}c(1-x^2)-1<x<1\\ 0\text{ otherwise}\end{array}$
(a) What is the value of c? (b) What is the cumulative distribution function of X?

The probability distribution of the random variable X represents the number of hits a baseball player obtained in a game for the 2012 baseball season.

$\begin{array}{|cc|}\hline x& P(x)\\ 0& 0.167\\ 1& 0.3289\\ 2& 0.2801\\ 3& 0.149\\ 4& 0.0382\\ 5& 0.0368\\ \hline\end{array}$
The probability distribution was used along with statistical software to simulate 25 repetitions of the experiment (25 games). The number of hits was recorded. Approximate the mean and standard deviation of the random variable X based on the simulation. The simulation was repeated by performing 50 repetitions of the experiment. Approximate the mean and standard deviation of the random variable. Compare your results to the theoretical mean and standard deviation. What property is being illustrated?
a) Compute the theoretical mean of the random variable X for the given probability distribution.
${\displaystyle \mu =?}$
b) Compute the theoretical standard deviation of the random variable X for the given probability distribution.
${\displaystyle \sigma =?}$
c) Approximate the mean of the random variable X based on the simulation for 25 games.
${\displaystyle \bar{X}=?}$
d) Approximate the standard deviation of the random variable X based on the simulation for 25 games.
${\displaystyle S=?}$

Suppose that the height, in inches, of a 25-year-old man is a normal random variable with parameters mu=71 and sigma^2=6.25 . What percentage of 25-year-old men are over 6 feet, 2 inches tall? What percentage of men in the 6-footer club are over 6 feet, 5 inches?

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.
$\begin{array}{|cccccccc|}\hline \text{Number of people}& 1& 2& 3& 4& 5& 6& 7\\ \begin{array}{l}\text{ Household }\text{ probability }\end{array}& 0.25& 0.32& 0.17& 0.15& 0.07& 0.03& 0.01\\ \begin{array}{l}\text{ Family }\text{ probability }\end{array}& 0& 0.42& 0.23& 0.21& 0.09& 0.03& 0.02\\ \hline\end{array}$
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.

In government data, a house-hold 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.
$\begin{array}{|cccccccc|}\hline \text{Number of people}& 1& 2& 3& 4& 5& 6& 7\\ \begin{array}{l}\text{ Household }\text{ probability }\end{array}& 0.25& 0.32& 0.17& 0.15& 0.07& 0.03& 0.01\\ \begin{array}{l}\text{ Family }\text{ probability }\end{array}& 0& 0.42& 0.23& 0.21& 0.09& 0.03& 0.02\\ \hline\end{array}$
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. The standard deviations of the 2 random variables are σH=1.421 and σF=1.249.. Explain why this difference makes sense.

Five people have a genetic disease and one child each. The random variable x is the number of children among the five who inherit the​ disease.
a)Find the mean.
b)Find the standard deviation.
$\begin{array}{|ccccccc|}\hline x& 0& 1& 2& 3& 4& 5\\ P(x)& 0.027& 0.162& 0.311& 0.311& 0.162& 0.027\\ \hline\end{array}$

Express the confidence interval ${\displaystyle 172.3<\mu <229.1}$ in the form of ${\displaystyle \bar{x}\pm ME.}$

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.

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
${\displaystyle M_x\left(t\right)=\frac{e^t(1-e^{kt})}{k(1-e^t)}}$.

Let X and Y be independent, continuous random variables with the same maginal probability density function, defined as
$f_X(t)=f_Y(t)=\{\begin{array}{l}\frac{2}{t^2},t>2\\ 0,otherwise\end{array}$
(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:
$f(x,y)=f(x,y)=\{\begin{array}{ll}c{x}^2{e}^{\frac{-y}{3}}& 1<x<6,2<y<4=0\\ 0& otherwise\end{array}$
Draw the integration boundaries and write the integration only for $P(X+Y\le 6)$