Exam 6: Random Variables and Discrete Probability Distributions

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 ?

Shopping Outlet A shopping outlet estimates the probability distribution of the number of stores shoppers actually enter as shown in the table below. {Shopping Outlet Narrative} What did you notice about the mean, variance, and standard deviation of Y = 2 X + 1 in terms of the mean, variance, and standard deviation of X ?

911 Phone Calls 911 phone calls arrive at the rate of 30 per hour at the local call center. {911 Phone Calls Narrative} Find the probability of receiving two calls in a five-minute interval of time.

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 the expected number of times a student may feel stressed during the final exam week.

The mean of a binomial distribution is equal to ____________________.

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. {Retries Narrative} What is the probability of no retries?

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 standard deviation of number of vehicles that are traveling on Montana highways and exceeding the speed limit.

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

A table, formula, or graph that shows all possible values a random variable can assume, together with their associated probabilities, is called a(n):

In the notation below, X is the random variable, E and V refer to the expected value and variance, respectively. Which of the following is false?

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

Let X be a Poisson random variable with m = 6. Use the table of Poisson probabilities to calculate: a. P ( X £ 8) b. P ( X = 8) c. P ( X ³ 5) d. P (6 £ X £ 10)

Classified Department Phone Calls A classified department receives an average of 10 telephone calls each afternoon between 2 and 4 P.M. The calls occur randomly and independently of one another. {Classified Department Phone Calls Narrative} Find the probability that the department will receive seven calls between 2 and 3 P.M. on a particular afternoon.

The Poisson random variable is a discrete random variable with infinitely many possible values.

Shopping Outlet A shopping outlet estimates the probability distribution of the number of stores shoppers actually enter as shown in the table below. x 0 1 2 3 4 p ( x )0.05 0.35 0.25 0.20 0.15 {Shopping Outlet Narrative} Suppose Y = 2 X + 1 for each value of X . What is the probability distribution of Y ?

Blackjack The probability distribution of a random variable X is shown below, where X represents the amount of money (in $1,000s)gained or lost in a particular game of Blackjack. {Blackjack Narrative} Find the following values, and indicate their units. a. E ( X ) b. V ( X ) c. Standard deviation of X

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:

The Poisson distribution is applied to events for which the probability of occurrence over a given span of time, space, or distance is very small.

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. {Number of Motorcycles Narrative} Apply the laws of expected value and variance to find the following: a. V( 3 X) b. V (3 X - 2) c. V( 3 ) d. V( 3 X)- 2

A remedial program evenly enrolls tradition and non-traditional students. If a random sample of 4 students is selected from the program to be interviewed about the introduction of a new on-line class, what is the probability that all 4 students selected are traditional students?