Probability Distributions for Discrete Random Variables. A discrete RV has a finite list of possible outcomes.

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1 1 Discrete Probability Distributions Section 4.1 Random Variable- its value is a numerical outcome of a random phenomenon. We use letters such as x or y to denote a random variable. Examples Toss a coin 10 times and record the number of heads Ask 100 people if they approve of the president and record the number of yes answers. Record scores on an exam for each student Count the number of defective items in a production line. Probability Distribution for a random variable x is the relative frequency distribution for the entire population; it shows what values of x occurred and how often these values occurred. Probability Distributions for Discrete Random Variables A discrete RV has a finite list of possible outcomes. The probability distribution of a discrete RV specifies all the values of x and their probabilities. We can make a probability histogram to show the probability distribution of a discrete random variable. The probabilities must satisfy the following two properties: o Every probability P(x) falls between 0 and 1. o All the probabilities sum to one. Parameters 1) Population Mean/ Expected Value: = E(X) = xp (x) 2 2 2) Population Standard Deviation: = ( x ) P( x) = E [( x ) ]

2 2 Example Grades in a very large statistics class are given according to the following distribution: A B C D F 15% 35% 30% 16%? a) Define the random variable X as the number of grade points given for each grade. Write the probability distribution of X. b) Is this a legitimate probability distribution? c) What is the probability that X is less than 3? d) What is the probability that X is less than or equal to three?

3 3 e) If there were exactly 100 students in the class, then 15 would have an A, 35 would have a B, 30 would have a C, 16 would have a D, and 4 would have an F. What is the expected grade for these 100 students?

4 4 Section 4.2 The Binomial Distribution: Probability for Counts with Binary Data The distribution of X = (number of successes) is called a Binomial Distribution with parameters n and p if it meets the following conditions: There are a fixed number of trials, n. The n trials are all independent. The outcome of each observation is either a success or a failure. Each trial has the same probability of success, p. The probability of failure (using compliment rule) is q = 1-p. Comment: The book uses P(S) for the probability of success. Notation: X ~ Bin (n, p) Examples Determine whether each of the following situations meet the criteria of a binomial distribution. If so, identify n and p. a) Toss a balanced coin n times and count the number of heads. b) Ask 100 randomly selected students if they drank when they were underage. Make sure that the questioning is handled in a way that the respondent s answer would be confidential. Count the number of students who answer yes to the question.

5 5 c) A shipment of 10,000 batteries arrives at a toy manufacturing company. To decide if they will purchase the whole shipment, a sample of 10 batteries is selected at random and tested. The number of defective batteries in the sample is counted. Unknown to the manufacturer, 1000 of the batteries in the shipment are defective. Population Proportion: p Sample Proportion: ^ p Population Mean: Sample Mean: X = np ^ n p Population Standard deviation: np ( 1 p) Sample Standard Deviation: s = n p(1 p) ^ ^ Calculating Probabilities for a Binomial Distribution P ) x n x ( x) ncx p (1 p for x = 0, 1,2,, n Where n! n C x and n! n( n 1)( n 2)...(2)(1) x!( n x)!

6 6 Example The juror pool for the upcoming murder trial of a celebrity actor contains the names of 100,000 individuals in the population who may be called for jury duty. The proportion of the available jurors on the population list who are Hispanic is.40. a jury of size 12 is selected at random from the population list of available jurors. Let X = the number of Hispanics selected to be jurors for this jury. a) Is it reasonable to assume that X has a binomial distribution? If not, why? If so, identify the values of n and p. b) Find the probability that no Hispanic is selected. c) How many jurors out of the 12 would you expect to be Hispanic? d) Find the standard deviation of this distribution.

7 7 Example A particular medication causes side effects on 35% of patients. Eight patients at our clinic are currently receiving that medication. Let X = number of these patients that experience side effects. a) Is the distribution of X binomial? b) Write down the sample space for this distribution. c) Find the probability that 6 out of the 8 patients experience side effects. d) How many patients out of 8 would you expect to experience side effects?

8 8 e) What is the probability that at most 3 patients experience side effects? f) What is the probability that at least 4 patients experience side effects? g) What is the probability that between 1 and 6 patients experience side effects (not inclusive)?

9 9 Help for Calculating Binomial Probabilities Calculate individual probabilities using the binomial formula for that particular value of x. Key words: exactly To calculate cumulative probabilities, sum the binomial formula results for each value of x. We will also look at how to use the Binomial Table. Key words: at least, at most, or more, or less. P ( X n) P( X n 1) = 1 P ( X n) P ( X n) P( X n 1) = 1 P ( X n) P ( m X n) P( X n) P( X m) = P( X n 1) P( X m ) for n > m

10 10 Section 4.3 The Poisson Distribution: Probability for the Number of Occurrences in an Event The distribution of X = (number of events that occur in an interval) is called a Poisson Distribution with parameter, where the average number of events that occur in an interval. Poisson distributions are: Unimodal Skewed Centered roughly at The spread increases as increases Notation: X ~ Pois ( ) Examples a) The number of deaths by horse kicking in the Prussian army. b) Cars pass through the I-275 and I-91interstate exchange at an average rate of 300 per hour. c) A life insurance salesman sells on the average 3 life insurance policies per week.

11 11 Poisson Distribution Population Mean: Sample Mean: ^ Population Standard deviation: Sample Standard Deviation: ^ = ^ Calculating Probabilities for a Poisson Distribution P( x) x e x! for x = 0, 1,2,, n Where x! x( x 1)( x 2)...(2)(1)

12 12 Example The births in a hospital occur randomly at an average rate of 1.8 births per hour. Let X = number of births in a given hour. a) What is the probability of observing 4 births in a given hour at the hospital? b) What is the probability of observing 2 or more births in a given hour at the hospital? c) What is the probability of observing 5 births in a given 2 hour interval?

13 13 Transformations with the Poisson Distribution

14 14 The Link Between the Binomial and Poisson Distribution When n is large and p is small, so that np<7, then a Bin(n, p) can be approximated with a Pois( ) where = np. In other words, under sufficient constraints on n and p: Bin(n, p) is approximately equivalent in distribution to a Pois( = np). Example Given that 5% of a population is left-handed, use the Poisson distribution to estimate the probability that a random sample of 100 people contains 2 or more left-handed people.

15 15 The Link Between the Binomial and Poisson Distribution When n is large and p is small, so that np<7, then a Bin(n, p) can be approximated with a Pois( ) where = np. In other words, under sufficient constraints on n and p: Bin(n, p) is approximately equivalent in distribution to a Pois( = np). Example: A company makes electric motors. The probability an electric motor is defective is.01. What is the probability that a sample of 300 electric motors will contain exactly 5 defective motors?

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