Exam 7: Random Variables and Discrete Probability Distributions

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} Compute the expected value of the portfolio composed of 60% stock 1 and 40% stock 2.

Which of the following is a discrete random variable?

Returns on Investment An analysis of the stock market produces the following information about the returns of two stocks. 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.

Elizabeth's Portfolio Elizabeth has decided to form a portfolio by putting 30% of her money into stock 1 and 70% into stock 2.She assumes that the expected returns will be 10% and 18%, respectively, and that the standard deviations will be 15% and 24%, respectively. -{Elizabeth's Portfolio Narrative} Compute the standard deviation of the returns on the portfolio assuming that the coefficient of correlation is 0.5.

Mobile Phones Sales After analyzing sales data, the owner of a Mobile Phone store produced the following joint probability distribution of the number of iPhones (X) and Blackberries (Y) sold daily. -{Mobile Phones Sales Narrative} Compute the expected number of iPhones sold daily.

Given a Poisson random variable X, where the average number of successes occurring in a specified interval is 1.8, then P(X = 0) is:

Post office The number of arrivals at a local post office between 3:00 and 5:00 P.M.has a Poisson distribution with a mean of 12. -{Post Office Narrative} Find the probability that the number of arrivals between 3:00 and 5:00 P.M.is at least 10.

For a random variable X, E(X + 2) - 5 = E(X) - 3, where E refers to the expected value.

The expected return of a portfolio of two investments will be equal to the sum of the expected returns of the two investments plus twice the covariance between the investments.

Mobile Phones Sales After analyzing sales data, the owner of a Mobile Phone store produced the following joint probability distribution of the number of iPhones (X) and Blackberries (Y) sold daily. -{Mobile Phones Sales Narrative} Find the marginal probability distribution of the number of Blackberries sold daily.

The time required to drive from New York to New Mexico is a discrete random variable.

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. -{Unsafe Levels of Radioactivity Narrative} Find the probability that there will be exactly 3 incidents in a year.

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} What is the distribution of X?

A continuous variable may take on any value within its relevant range even though the measurement device may not be precise enough to record it.

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 maximizing her returns, which stock should she choose?

Mobile Phones Sales After analyzing sales data, the owner of a Mobile Phone store produced the following joint probability distribution of the number of iPhones (X) and Blackberries (Y) sold daily. -{Mobile Phones Sales Narrative} Find the probability distribution of X + Y.

Number of Birds Alana and Eva are sisters.Let X denote the number of birds that Alana may have in the next two years, and let Y denote the number of birds Eva may have during the same period.The marginal probability distributions of X and Y are shown below. -{Number of Birds Narrative} Verify that E(XY) = E(X)E(Y).Did you expect this result? Why?

The number of accidents that occur annually on a busy stretch of highway is an example of:

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.

Number of Birds Alana and Eva are sisters.Let X denote the number of birds that Alana may have in the next two years, and let Y denote the number of birds Eva may have during the same period.The marginal probability distributions of X and Y are shown below. -{Number of Birds Narrative} Assume that X and Y are independent and find their bivariate distribution.