Exam 6: Random Variables and Discrete Probability Distributions

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 P (2 £ X £ 4).

Determine which of the following are not valid probability distributions, and explain why not.

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} Find the following probabilities:

The number of customers arriving at a department store in a 5-minute period has a Poisson distribution.

The number of accidents that occur at a busy intersection in one month is an example of a Poisson random variable.

The binomial probability distribution is a discrete probability distribution.

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 P ( X = 10).

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 variance for the number of retries?

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 variance and standard deviation.

A function or rule that assigns a numerical value to each simple event of an experiment is called:

The trials in a binomial experiment are ____________________, meaning the outcome of one trial does not affect the outcomes of any other trials.

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 probabilities: a. P ( X £ 0) b. P ( X > 3) c. P (0 £ X £ 4) d. P ( X = 5)

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 )= 0.05 x , x = 2, 3, 4, 5, or 6. {Number of Horses Narrative} Find the expected number of horses per family.

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} Calculate the variance and standard deviation of Y directly from the probability distribution of Y .

If the probability of success p remains constant in a binomial distribution, an increase in n will increase the variance.

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.

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.

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. Given a binomial random variable with n = 20 and p = 0.60, find the following probabilities using the binomial table.

The largest value that a Poisson random variable X can have is n .

An auto insurance company evaluates many numerical variables about a person before deciding on an appropriate rate for automobile insurance. A person's age is an example of a(n)____________________ random variable.