Exam 7: A: Random Variables and Discrete Probability Distributions

Car Sales The joint probability distribution of variables X and Y is shown in the table below.Rebecca and Rachel are car salespeople.Let X denote the number of cars that Rebecca will sell in a month,and let Y denote the number of cars Rachel will sell in a month. $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ X 1 2 3 1 0.30 0.18 0.12 2 0.15 0.09 0.06 3 0.05 0.03 0.02 -{Car Sales Narrative} Calculate V(X)and V(Y).

Number of Horses The random variable X represents the number of horses per family in a rural area in Iowa,with the probability distribution: p(x)= 0.05x,x = 2,3,4,5,or 6. -{Number of Horses Narrative} Find the expected number of horses per family.

Retries The following table contains the probability distribution for X = the number of retries necessary to successfully transmit a 1024K data package through a double satellite media. 0 1 2 3 ( ) 0.35 0.35 0.25 0.05 -If X and Y are independent variables,then COV(X,Y)> 0.

If the probability of success p remains constant in a binomial distribution,an increase in n will not change the mean.

Returns on Investment An analysis of the stock market produces the following information about the returns of two stocks. Stock 1 Stok 2 Bxpseted Returns 15\% 18\% Standand Deviations 20 32 Assume that the returns are positively correlated with correlation coefficient of 0.80. -{Returns on Investment Narrative} Suppose that you wish to invest $1 million.Discuss whether you should invest your money in stock 1,stock 2,or a portfolio composed of an equal amount of investments on both stocks.

Golfing Store The joint probability distribution of variables X and Y is shown in the table below,where X is the number of drivers and Y is the number of putters sold daily in a small golfing store. $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ X 1 2 3 1 0.30 0.18 0.12 2 0.15 0.09 0.06 3 0.05 0.03 0.02 -{Golfing Store Narrative} Verify that V(X + Y)= V(X)+ V(Y).Did you expect this result? Why?

Gym Visits Let X represent the number of times a student visits a gym in a one month period.Assume that the probability distribution of X is as follows: 0 1 2 3 ( ) 0.05 0.25 0.50 0.20 -{Gym Visits Narrative} What is the probability that the student visits the gym at least once in a month?

Golfing Store The joint probability distribution of variables X and Y is shown in the table below,where X is the number of drivers and Y is the number of putters sold daily in a small golfing store. $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ X 1 2 3 1 0.30 0.18 0.12 2 0.15 0.09 0.06 3 0.05 0.03 0.02 -{Golfing Store Narrative} Find P(Y = 2 | X = 1)

Number of Motorcycles The probability distribution of a discrete random variable X is shown below,where X represents the number of motorcycles owned by a family. 0 1 2 3 ( ) 0.25 0.40 0.20 0.15 -{Number of Motorcycles Narrative} Find the following probabilities: a. P(X > 1) b. P(X $\le$ 2) c. P(1 $\le$ X $\le$ 2) d. P(0 < X < 1) e. P(1 $\le$ X < 3)

Katie's Portfolio Katie is given the following information about the returns on two stocks: E(R₁)= 0.10,E(R₂)= 0.15,V(R₁)= 0.0225,and V(R₂)= 0.0441. -{Katie's Portfolio Narrative} If Katie is most interested in minimizing her risk,which stock should she choose?

Gym Visits Let X represent the number of times a student visits a gym in a one month period.Assume that the probability distribution of X is as follows: 0 1 2 3 ( ) 0.05 0.25 0.50 0.20 -{Gym Visits Narrative} Find the mean and the standard deviation of Y = 2X - 1.

A(n)____________________ random variable is one whose values are countable.

The expected return of a two-asset portfolio is equal to the product of the weight assigned to the first asset and the expected return of the first asset plus the product of the weight assigned to the second asset and the expected return of the second asset.

Montana Highways A recent survey in Montana revealed that 60% of the vehicles traveling on highways,where speed limits are posted at 70 miles per hour,were exceeding the limit.Suppose you randomly record the speeds of ten vehicles traveling on US 131 where the speed limit is 70 miles per hour.Let X denote the number of vehicles that were exceeding the limit. -{Montana Highways Narrative} Find the expected number of vehicles that are traveling on Montana highways and exceeding the speed limit.

A function or rule that assigns a numerical value to each outcome of an experiment is called:

Unsafe Levels of Radioactivity The number of incidents at a nuclear power plant has a Poisson distribution with a mean of 6 incidents per year. -The amount of milk consumed by a baby in a day is an example of a discrete random variable.

Stress Consider a binomial random variable X with n = 5 and p = 0.40,where X represents the number of times in the final exam week a student with 18 credit hours may feel stressed. -{Stress Narrative} Find P(2 $\le$ X $\le$ 4).

Shopping Outlet A shopping outlet estimates the probability distribution of the number of stores shoppers actually enter as shown in the table below. 0 1 2 3 4 ( ) 0.05 0.35 0.25 0.20 0.15 -{Shopping Outlet Narrative} Find the expected value of the number of stores entered.

Golfing Store The joint probability distribution of variables X and Y is shown in the table below,where X is the number of drivers and Y is the number of putters sold daily in a small golfing store. $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ X 1 2 3 1 0.30 0.18 0.12 2 0.15 0.09 0.06 3 0.05 0.03 0.02 -{Golfing Store Narrative} Determine the marginal probability distributions of X and Y.

Which of the following is not a required condition for the distribution of a discrete random variable X that can assume values x_i?