Exam 7: Random Variables and Discrete Probability Distributions

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} Develop the probability distribution of X + Y.

One of the ways in which financial analysts lower the risk that is associated with the stock market is through diversification.

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 -If X has a binomial distribution with n = 4 and p = 0.3, find the probability that X is at most one.

NARRBEGIN: Stress 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.NARREND -{Stress Narrative} Find the variance and standard deviation.

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} Find the probability that at least 1 student is a sports fan.

The amount of milk consumed by a baby in a day is an example of a discrete random variable.

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} If Katie is most interested in minimizing her risk, which stock should she choose?

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} Calculate E(X) and E(Y).

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 variance to calculate the variance and standard deviation of Y from the probability distribution of X.

In the notation below, X is the random variable, c is a constant, and V refers to the variance. Which of the following laws of variance is not true?

The Poisson random variable is a:

The variance of the sum always equals the sum of the variances.

NARRBEGIN: Gym Visits 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 NARREND -The monthly sales at a Gas Station have a mean of $50,000 and a standard deviation of $6,000. Profits are calculated by multiplying sales by 40% and subtracting fixed costs of $12,000. Find the mean and standard deviation of monthly profits.

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} Apply the laws of expected value to find the following: a.E(X²) b.E(2X² + 5) c.E(X - 2)²

Evaluate the following binomial coefficients. a. b. c. d.

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} Determine the marginal probability distribution of Y.

NARRBEGIN: Stress 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.NARREND -{Stress Narrative} Find the expected number of times a student may feel stressed during the final exam week.

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 coefficient of correlation between X and Y. Did you expect this result? Why?

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} Determine the marginal probability distribution of X.

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 -{Number of Horses Narrative} Find the following probabilities: a.P(X $\ge$ 4) b.P(X > 4) c.P(3 $\le$ X $\le$ 5) d.P(2 < X < 4) e.P(X = 4.5)