# Random variables problems solved # Recent questions in Random variables

Random variables
ANSWERED ### 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.

Random variables
ANSWERED ### 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.

Random variables
ANSWERED ### 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 $$\displaystyle{M}_{{{x}}}{\left({t}\right)}={\frac{{{e}^{{{t}}}{\left({1}-{e}^{{{k}{t}}}\right)}}}{{{k}{\left({1}-{e}^{{{t}}}\right)}}}}$$.

Random variables
ANSWERED ### 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? $$\begin{array}{|c|c|}\hline & \text{No} & \text{Yes} \\ \hline \text{Gender} \\ \hline \text{Female} & 15 & 22 \\ \hline \text{Male} & 12 & 32 \\ \hline \end{array}$$

Random variables
ANSWERED ### 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).

Random variables
ANSWERED ### Chances: Risk and Odds in Everyday Life, by James Burke, reports that only 2% of all local franchises are business failures. A Colorado Springs shopping complex has 137 franchises (restaurants, print shops, convenience stores, hair salons, etc.). Let r be the number of these franchises that are business failures. Explain why a Poisson approximation to the binomial would be appropriate for the random variable r. What is n? What is p? What is λλ (rounded to the nearest tenth)?

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

Random variables
ANSWERED ### 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.

Random variables
ANSWERED ### Two random variables X and Y with joint density function given by: $$f(x,y)=\begin{cases}\frac{1}{3}(2x+4y)& 0\leq x,\leq 1\\0 & elsewhere\end{cases}$$ Find the marginal density of X.

Random variables
ANSWERED ### 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)$$

Random variables
ANSWERED ### Find the correlation coefficient between the tandom variables X and Y if the covariance of X and Y is 0.9 and variances of the random variables X and Y are 4 and 9, respectively 1)0.142 2)0.133 3)0.150 4)0.125

Random variables
ANSWERED ### Let $$X_{1}, X_{2},...,X_{n}$$ be n independent random variables each with mean 100 and standard deviation 30. Let X be the sum of these random variables. Find n such that $$Pr(X>2000)\geq 0.95$$.

Random variables
ANSWERED ### 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.

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

Random variables
ANSWERED ### 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?

Random variables
ANSWERED ### Assume that X and Y are jointly continuous random variables with joint probability density function given by $$f(x,y)=\begin{cases}\frac{1}{36}(3x-xy+4y)\ if\ 0 < x < 2\ and\ 1 < y < 3\\0\ \ \ \ \ othrewise\end{cases}$$ Find the marginal density functions for X and Y .

Random variables
ANSWERED ### Find the expected values and variances of the exponential random variables with the density functions given below $$\frac{1}{4}e^{-\frac{x}{4}}$$

Random variables
ANSWERED ### The joint density of the random variables X and Y is given by $$f(x,y)=\begin{cases}8xy & 0\leq x\leq 1, 0\leq y \leq x\\0 & otherwise\end{cases}$$ Find the marginal density of X

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