Exam 7: Random Variables and Discrete Probability Distributions

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} Find the marginal probability distribution of the number of Blackberries sold daily.

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} Calculate E(X + Y) directly by using the probability distribution of X + Y.

NARRBEGIN: Unsafe Levels of Radioact 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.NARREND -{Unsafe Levels of Radioactivity Narrative} Find the probability that there will be at least 3 incidents in a year.

The Sutton police department must write, on average, 6 tickets a day to keep department revenues at budgeted levels. Suppose the number of tickets written per day follows a Poisson distribution with a mean of 6.5 tickets per day. Interpret the value of the mean.

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} Find the expected mean of the portfolio.

If X and Y are independent random variables, which of the following identities is false?

In Poisson experiment, the probability of more than one success in an interval approaches ____________________ as the interval becomes smaller.

The variance of a binomial distribution for which n = 50 and p = 0.20 is 8.0.

A random variable is a function or rule that assigns a number to each outcome of an experiment.

The covariance between two investments of a portfolio is equal to the sum of the variances of the investments.

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} Verify that E(XY) = E(X)E(Y). Did you expect this result? Why?

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} Compute the mean and variance of Y.

An auto insurance company evaluates many numerical variables about a person before deciding on an appropriate rate for automobile insurance. The number of claims a person has made in the last 3 years is an example of a(n) ____________________ 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} Calculate the expected value of Y directly from the probability distribution of Y.

If X and Y are any random variables with COV(X, Y) = 0.25, $\sigma _ { x } ^ { 2 } = 0.36$ , and $\sigma _ { y } ^ { 2 } = 0.49$ , then the coefficient of correlation $\rho$ is

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} Find the probability distribution of X + Y.

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} Compute the covariance between X and Y.

If X and Y are independent variables, then COV(X, Y) > 0.

NARRBEGIN: Number of Horses 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.NARREND -Determine which of the following are not valid probability distributions, and explain why not. a.x 0 1 2 3 p(x) 0.15 0.25 0.35 0.45 b.x 2 3 4 5 p(x) -0.10 0.40 0.50 0.25 c.x -2 -1 0 1 2 p(x) 0.10 0.20 0.40 0.20 0.10

NARRBEGIN: Sports Fans Sports Fans Suppose that past history shows that 5% of college students are sports fans. A sample of 10 students is to be selected.NARREND -{Sports Fans Narrative} A sample of 100 students is to be selected. What is the average number that you would expect to sports fan?