Discrete Probability Distributions Martina Litschmannová K210

 A random variable is a function or rule that assigns a number to each outcome of an experiment.  Basically it is just a symbol that represents the outcome of an experiment. Random Variable

Discrete Random Variable  usually count data [Number of]  one that takes on a countable number of values – this means you can sit down and list all possible outcomes without missing any, although it might take you an infinite amount of time. For example:  X = values on the roll of two dice: X has to be either 2, 3, 4, …, or 12.  Y = number of accidents in Ostrava during a week: Y has to be 0, 1, 2, 3, 4, 5, 6, 7, 8, ……………”real big number”

Binomial Experiment

Binomial Experiment - example You flip a coin 5 times and count the number of times the coin lands on heads. This is a binomial experiment because:  The experiment consists of repeated trials. We flip a coin 5 times.  Each trial can result in just two possible outcomes - heads or tails.  The probability of success is constant – 0,5 on every trial.  The trials are independent. That is, getting heads on one trial does not affect whether we get heads on other trials.

Binomial Distribution # of trialsprobability of succeses

1.Suppose a die is tossed 5 times. What is the probability of getting exactly 2 fours?

OutcomeProbability SSFFF SFSFF SFFSF SFFFS FSSFF FSFSF FSFFS FFSSF FFSFS FFFSS 1.Suppose a die is tossed 5 times. What is the probability of getting exactly 2 fours?

OutcomeProbability SSFFF SFSFF SFFSF SFFFS FSSFF FSFSF FSFFS FFSSF FFSFS FFFSS 1.Suppose a die is tossed 5 times. What is the probability of getting exactly 2 fours?

2.If the probability of being a smoker among a group of cases with lung cancer is 0,6, what’s the probability that in a group of 80 cases you have: a) less than 20 smokers, b) more than 50 smokers, c) greather than 10 and less than 40 smokers? d) What are the expected value and variance of the number of smokers?

Negative Binomial Experiment

Negative Binomial Experiment - example You flip a coin repeatedly and count the number of times the coin lands on heads. You continue flipping the coin until it has landed 5 times on heads. This is a negative binomial experiment because:  The experiment consists of repeated trials. We flip a coin repeatedly until it has landed 5 times on heads.  Each trial can result in just two possible outcomes - heads or tails.  The probability of success is constant – 0,5 on every trial.  The trials are independent. That is, getting heads on one trial does not affect whether we get heads on other trials.  The experiment continues until a fixed number of successes have occurred; in this case, 5 heads.

Negative Binomial Distribution (Pascal Distribution)

Geometric Distribution

3.Bob is a high school basketball player. He is a 70% free throw shooter. That means his probability of making a free throw is 0,70. During the season, what is the probability that Bob makes his third free throw on his fifth shot?

4.Bob is a high school basketball player. He is a 70% free throw shooter. That means his probability of making a free throw is 0,70. During the season, what is the probability that Bob makes his first free throw on his fifth shot?

Hypergeometric Experiments A hypergeometric experiment is a statistical experiment that has the following properties:  A sample of size n is randomly selected without replacement from a population of N items.  In the population, M items can be classified as successes, and N - M items can be classified as failures.

Hypergeometric Experiment - example You have an urn of 10 balls - 6 red and 4 green. You randomly select 2 balls without replacement and count the number of red balls you have selected. This would be a hypergeometric experiment. N M successes N-M failures k selected items

Hypergeometric Distribution

5.Suppose we randomly select 5 cards without replacement from an ordinary deck of playing cards. What is the probability of getting exactly 2 red cards (i.e., hearts or diamonds)? N=52 M=26 N-M=26

Poisson Experiment A Poisson experiment is a statistical experiment that has the following properties:  The experiment results in outcomes that can be classified as successes or failures.  The average number of successes (μ) that occurs in a specified region is known.  The probability that a success will occur is proportional to the size of the region.  The probability that a success will occur in an extremely small region is virtually zero. Note that the specified region could take many forms. For instance, it could be a length, an area, a volume, a period of time, etc.

Poisson Distribution

6.The average number of homes sold by the Acme Realty company is 2 homes per day. What is the probability that less than 4 homes will be sold tomorrow?

7.Suppose the average number of lions seen on a 1-day safari is 5. What is the probability that tourists will see fewer than twelve lions on the next 3-day safari?

Study materials :  (p p.79)  (Distributions - Discrete)