Exam 5: Discrete Random Variables

Suppose that the random variable X has a binomial distribution and that the success probability, p, is greater than 0.5. Is the probability distribution of X right skewed, left skewed, or symmetric? Explain your thinking.

Five cards are drawn at random, with replacement, from an ordinary deck of 52 cards. Considering success to be drawing a heart, formulate the process of observing the suits of the five cards as a sequence of five Bernoulli trials.

Use the Poisson Distribution to find the indicated probability. Round to three decimal places when necessary. -The number of calls received by a car towing service in an hour has a Poisson distribution with parameter $\lambda = 1.64$ . Find the probability that in a randomly selected hour the number of calls is between 2 and 4 inclusive.

Explain in your own words the meaning of the term "probability distribution".

Determine the possible values of the random variable. -Suppose that two balanced dice are rolled. Let X denote the absolute value of the difference of the two numbers. What are the possible values of the random variable X?

Find the expected value of the random variable. Round to the nearest cent unless stated otherwise. -In a game, you have a 1/33 probability of winning $62 and a 32/33 probability of losing $2. What is your expected value?

Find the mean of the binomial random variable. Round to two decimal places when necessary. -In a certain town, 90 percent of voters are in favor of a given ballot measure and 10 percent are opposed. For groups of 180 voters, find the mean for the random variable X, the number who Oppose the measure.

Provide an appropriate response. -A bag contains 4 red marbles and 6 green marbles. Anne picks 3 marbles at random, with replacement, and observes the color of each marble. The number of green marbles, X, is a binomial Random variable. If we let success correspond to getting a green marble, what is the success Probability, p? What is the number of trials?

Construct a probability histogram for the binomial random variable, X. -Three coins are tossed. X is the number of tails.

A coin is biased. Danny wishes to determine the probability of obtaining heads when flipping this coin. He flips the coin 10 times and obtains 8 heads. He concludes that the probability of obtaining heads when flipping this coin is 0.8. Is his thinking reasonable? Why or why not?

Find the expected value of the random variable. Round to the nearest cent unless stated otherwise. -The probability distribution below describes the number of thunderstorms that a certain town may experience during the month of August. Let X represent the number of thunderstorms in August. Number of storms 0 1 2 3 (X=) 0.1 0.2 0.5 0.2 What is the expected value of thunderstorms for the town each August? Round the answer to one Decimal place.

Use random-variable notation to represent the event. -Suppose that two balanced dice are rolled. Let Y denote the sum of the two numbers. Use random-variable notation to represent the event that the sum of the two numbers is at least 11.

Find the specified probability. -A statistics professor has office hours from 9:00 am to 10:00 am each day. The number of students waiting to see the professor is a random variable, X, with the distribution shown in the table. 0 1 2 3 4 5 (=) 0.05 0.10 0.40 0.25 0.15 0.05 The professor gives each student 10 minutes. Determine the probability that a student arriving just After 9:00 am will have to wait at least 30 minutes to see the professor.

6.2% of VCRs of a certain type are defective. Let the random variable X represent the number of defective VCRs among 200 randomly selected VCRs of this type. Suppose you wish to find the probability that X is equal to 8. Does the random variable X have a binomial or a Poisson distribution? How can you tell? If X has a binomial distribution, would it be reasonable to use the Poisson approximation? If not, why not?

Identify each of the variables in the Binomial Probability Formula. $P ( x ) = \frac { n ! } { ( n - x ) ! x ! } \cdot p ^ { x } \cdot ( 1 - p ) ^ { n - x }$ Also, explain what the fraction $\frac { n ! } { ( n - x ) ! x ! }$ computes.

Obtain the probability distribution of the random variable. -The following frequency table contains data on home sale prices in the city of Summerhill for the month of June. For a randomly selected sale price between $80,000 and $265,900 let X denote the Number of homes that sold for that price. Find the probability distribution of X. Sale Price (in thousands) Frequency (No. of homes sold) 80.0-110.9 2 111.0-141.9 5 142.0-172.9 7 173.0-203.9 10 204.0-234.9 3 235.0-265.9 1

If the random variable X has a Poisson distribution, then the probability distribution of X can be either right skewed or symmetric?

Find the standard deviation of the random variable. -A police department reports that the probabilities that 0, 1, 2, and 3 burglaries will be reported in a given day are 0.48, 0.36, 0.14, and 0.02, respectively. Find the standard deviation for the probability Distribution. Round the answer to two decimal places.

The number of lightning strikes in a year at the top of a particular mountain has a Poisson distribution with parameter $\lambda = 3.5$ . Construct a histogram of the probabilities when the number of strikes is from 1-5.

Use random-variable notation to represent the event. -The following table displays a frequency distribution for the number of siblings for students in one middle school. For a randomly selected student in the school, let X denote the number of siblings of The student. Number of siblings 0 1 2 3 4 5 6 7 Frequency 189 245 102 42 24 13 5 2 Use random-variable notation to represent the event that the student obtained has fewer than two siblings.