Chapter 7 Special Discrete Distributions

Binomial Distribution Each trial has two mutually exclusive possible outcomes: success/failure Fixed number of trials (n) Trials are independent Probability of success (p) is the same for all trials Binomial random variable: X = the number of successes

Are these binomial distributions? 1)Toss a coin 10 times and count the number of heads Yes 2)Deal 10 cards from a shuffled deck and count the number of red cards No, probability of red does not remain the same 3)Doctors at a hospital note whether babies born to mothers with type O blood also have type O blood No, number of trials isn't fixed

Toss a 3 coins and count the number of heads Construct the discrete probability distribution. x P(x) Out of 3 coins that are tossed, what is the probability of getting exactly 2 heads?

Binomial Formula: Where:

Out of 3 coins that are tossed, what is the probability of getting exactly 2 heads?

The number of inaccurate pistons in a group of four is a binomial random variable. If the probability of a defect is 0.1, what is the probability that only 1 is defective? More than 1 is defective?

Calculator binompdf(n, p, x)  P(X = x) binomcdf(n, p, x)  P(X < x)  Cumulative probabilities from P(0) to P(x)

A genetic trait in one family manifests itself in 25% of the offspring. If eight offspring are randomly selected, find the probability that the trait will appear in exactly three of them. At least five of them?

In a certain county, 30% of the voters are Democrats. If ten voters are selected at random, find the probability that no more than six of them will be Democrats. P(X < 6) = binomcdf(10,.3,6) =.9894 not What is the probability that at least 7 are not Democrats? P(X > 7) = 1 – binomcdf(10,.7,6) =.6496

Skewed right  Symmetrical at p =.5  Skewed left What happened to the shape of the distribution as the probability of success increased?

What do you notice about the means and standard deviations? As p increases, the means increase the standard deviations increase until p =.5, then decrease

Binomial Mean and Standard Deviation

In a certain county, 30% of the voters are Democrats. How many Democrats would you expect in ten randomly selected voters? What is the standard deviation for this distribution? expect

Geometric Distribution Two mutually exclusive outcomes Each trial is independent Probability of success remains constant Random variable: X = number of trials UNTIL the FIRST success So what are the possible values of X? X 1234 How far will this go?... To infinity

Geometric: NOT a fixed number of trials  no "n" Binomial starts with 0; Geometric starts with 1 Binomial dist.: finite; Geometric dist.: infinite Differences between Binomial & Geometric

Count the number of boys in a family of four children. Binomial: X01234 Count children until first son is born Geometric:... X

Geometric Formulas Not on green sheet – they will be given if needed on a test

Calculator P(X = x) = geometpdf(p, x) P( X < x) = geometcdf(p, x)  Cumulative probability from 1 to x No “n” because there is no fixed number of trials

What is the probability that the first son is the fourth child born? What is the probability that the first son is born in at most four children?

A real estate agent shows a house to prospective buyers. The probability that the house will be sold is 35%. What is the probability that the agent will sell the house to the third person she shows it to? How many prospective buyers does she expect to show the house to before someone buys the house?

Poisson Distribution Deals with infrequent events Examples: Accidents per month at an intersection Tardies per semester for a student Runs per inning in a baseball game

Properties A discrete number of events occur in a continuous interval Each interval is independent of other intervals P(success) in an interval is the same for all intervals of equal size P(success) is proportional to the size of the interval

Formulas X = # of events per unit of time, space, etc. λ (lambda) = mean of X

The average number of accidents in an office building during a four-week period is 2. What is the probability that there will be one accident in the next four-week period? What is the probability that there will be more than two accidents in the next four- week period?

The number of calls to a police department between 8 pm and 8:30 pm on Friday averages 3.5. What is the probability of no calls during this period? What is the probability of no calls between 8 pm and 9 pm on Friday night? What is the mean and standard deviation of the number of calls between 10 pm and midnight on Friday night? P(X = 0) = poissonpdf(3.5, 0) =.0302 P(X = 0) = poissonpdf(7, 0) = :00 to 8:30 is a 30 minute interval. 8:00 to 9:00 is a 60 minute interval. Since the interval is doubled, we double the mean amount of calls to keep it proportional. μ = 14 & σ = Be sure to adjust λ!

Let's examine histograms of the Poisson distribution. λ = 2 λ = 4 λ = 6 What happens to the shape? What happens to the mean? What happens to the standard deviation?

As λ increases, Distribution becomes more symmetrical Mean and standard deviation both increase