Exam 7: Random Variables and Discrete Probability Distributions

NARRBEGIN: Number of Birds 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. 0 1 2 ( ) 0.5 0.3 0.2 y 0 1 2 ( ) 0.4 0.5 0.1 NARREND -{Number of Birds Narrative} Calculate V(X + Y) directly by using the probability distribution of X + Y.

NARRBEGIN: Golfing Store 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. 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 NARREND -{Golfing Store Narrative} Determine the marginal probability distributions of X and Y.

If X and Y are independent, then the coefficient of correlation equals ____________________.

NARRBEGIN: Online Bankers Online Bankers An official from the securities commission estimates that 75% of all online bankers have profited from the use of insider information. Assume that 15 online bankers are selected at random from the commission's registry.NARREND -{Online Bankers Narrative} Find the probability that all 15 have profited from insider information.

NARRBEGIN: Mobile Phones Sales 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. X 1 1 0.4 0.1 2 0.3 0.2 NARREND -{Mobile Phones Sales Narrative} Compute the variance of the number of iPhones sold daily.

NARRBEGIN: Elizabeth's Portfolio 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.NARREND -{Elizabeth's Portfolio Narrative} Compute the standard deviation of the returns on the portfolio assuming that the two stocks' returns are uncorrelated.

NARRBEGIN: Shopping Outlet 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 \square.20 0.15 NARREND -{Shopping Outlet Narrative} Use the laws of expected value to calculate the mean of Y from the probability distribution of X.

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

NARRBEGIN: Shopping Outlet 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 \square.20 0.15 NARREND -{Shopping Outlet Narrative} What did you notice about the mean, variance, and standard deviation of Y = 2X + 1 in terms of the mean, variance, and standard deviation of X?

NARRBEGIN: Number of Motorcycles 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 NARREND -{Number of Motorcycles Narrative} Find the expected value of X.

NARRBEGIN: Golfing Store 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. 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 NARREND -{Golfing Store Narrative} Find the probability distribution of the random variable X + Y.

NARRBEGIN: Montana Highways 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.NARREND -{Montana Highways Narrative} Find the expected number of vehicles that are traveling on Montana highways and exceeding the speed limit.

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.

A Poisson random variable is a(n) ____________________ random variable.

A table, formula, or graph that shows all possible values a random variable can assume, together with their associated probabilities, is referred to as discrete probability distribution.

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

NARRBEGIN: Retries 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 NARREND -{Retries Narrative} What is the mean or expected value for the number of retries?

In the Poisson distribution, the mean is equal to the ____________________.

NARRBEGIN: Car Sales 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. 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 NARREND -{Car Sales Narrative} Verify that E(X + Y) = E(X) + E(Y).

NARRBEGIN: Katie's Portfolio 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.NARREND -{Katie's Portfolio Narrative} Compute the variance of the portfolio composed of 60% stock 1, and 40% stock 2, if the coefficient of correlation is 0.40.