1 Chapter 3 Discrete Random Variables and Probability Distributions Presenting the Theoretical Distributions Uniform Binomial Geometric Poisson Chapter 3B ENM 500 campus students excited about today’s lecture

2 What You Should Know about the Upcoming Discrete Distributions  The type of stochastic situations that give rise to the distribution. What is the ‘classic’ scenario? What distinguishes this distribution’s application from the others?  The type of real world situations you can model with these distributions.  How do you turn the crank and generate some valid probabilities given a scenario description?

3 Discrete Uniform Distribution Definition: a random variable X is a discrete uniform random variable if each of the n values in its range {x 1, x 2,….x n } has equal probability. Then f(x i ) = 1/n “Discrete” implies that only specific values within the range are possible. For example, a digital weight scale reads between 0 and 10 pounds and has two decimal place accuracy. The possible values are {0.00, 0.01, 0.02, …, 10.00}. Other values, such as are in the range but are not possible values.

4 3-5 Discrete Uniform Distribution Definition

5 3-5 Discrete Uniform Distribution Example 3-13

6 3-5 Discrete Uniform Distribution Figure 3-7 Probability mass function for a discrete uniform random variable.

7 3-5 Discrete Uniform Distribution Mean and Variance “follows” parameters

8 Finding the Mean The sum of an arithmetic series where a is the first term, n is the number of terms, and d= is the difference between consecutive values.

9 A Uniform Example  The number of demands per day for a symmetrical, spiral, closed-ended sprocket is

Binomial Distribution Random experiments and random variables

Binomial Distribution Random experiments and random variables

12 Binomial Distribution Many random experiments produce results that fall into patterns. One such pattern can be characterized as follows:  A series of independent random trials occurs  Each trial can be summarized as a “success” or a “failure” – called a Bernoulli trial  The probability of success on each trial (p) remains constant  The random variable of interest is a count of the number of successes in n trials. Independent Bernoulli trials

13 Jacob Bernoulli BornDecember 27December 27, 1654) Basel, Switzerland1654 BaselSwitzerland DiedAugust 16August 16, 1705 (aged 50) Basel, Switzerland1705 BaselSwitzerland Nationality Swiss FieldMathematician InstitutionsUniversity of Basel Academic advisor Gottfried Leibniz Notable students Johann Bernoulli Jacob Hermann Nicolaus I Bernoulli Known forBernoulli trial Bernoulli numbers ReligionCalvinist

Binomial Distribution Definition

15 Deriving the PMF Let n = 7 and p =.2 X = success 0 = failure X 0 0 X 0 X 0 This is just the number of permutations of 7 objects where 3 are the same and the other 4 are the same.

16 Is this a PMF? Binomial Theorem:

Binomial Distribution Figure 3-8 Binomial distributions for selected values of n and p.

Binomial Distribution Example 3-18

Binomial Distribution Example 3-18

20 Mean & Variance of a Binomial Distribution Let X = number of successes in n trials What value of p gives the largest variance or uncertainty?

Binomial Distribution Mean and Variance

22 Problem 3-79 (overbooking) Assume 120 seats are available on an airliner, 125 tickets were sold. The probability a passenger does not show is X = actual number of no- shows.  What is the probability that every passenger who shows up can take this flight?  What is the probability that the flight departs with empty seats?

23 A Binomial Example  The probability of a forty-year old male dying in his 40 th year is A group of 20 forty-year old males meet at their high school reunion. What is the probability that at least one of them will be dead before the year is over? Let X = a discrete random variable, the number of deaths among year old males. X = 0, 1, 2, …, 20. Pr{X  1} = 1 – f(0) = =.0505

Geometric Distributions Definition sample space S={s, fs, ffs, fffs, …}

Geometric Distributions Figure 3-9. Geometric distributions for selected values of the parameter p.

Geometric Distributions Geometric Distribution Example 3-21

Geometric Distributions The overachieving student may wish to fill in the gaps

28 Geometric Distribution Geometric series p = a r = 1 -p

29 A Geometric example  There is one chance in 10,000 of winning a particular lottery game. How many games must be played to achieve the first win? Let X = a discrete random variable, the number of games played to achieve the first win.

Geometric Distributions Lack of Memory Property

Poisson Distribution Definition

32 Siméon Denis Poisson  In Recherches sur la probabilité des jugements en matière criminelle et matière civile, an important work on probability published in 1837, the Poisson distribution first appears. The Poisson distribution describes the probability that a random event will occur in a time or space interval under the conditions that the probability of the event occurring is very small, but the number of trials is very large so that the event actually occurs a few times.probability Born: 1781 in Pithiviers, France Died: in Sceaux, France

33 Situations That Lead to the Poisson  Telephone calls arriving at a help desk  Number of alpha particles emitted from a radioactive source  Number of accidents occurring over a given time period (used by insurance industry)  Other examples are customers arriving at a store, bank, or fast food outlet.  In light traffic, the number of vehicles that pass a marker on a roadway  The arrival of natural events such as tornadoes, hurricanes, and lightning strikes.

34 Is this a PMF? Why it really is a probability mass function. recall:

Poisson Distribution Mean and Variance The overachieving student may wish to derive the variance

Poisson Distribution Consistent Units

Poisson Distribution Example 3-33

Poisson Distribution Example 3-33

39 Problem Telephone calls arrive at a phone exchange with Poisson Distribution; = 10 calls / hour What is the probability there are 3 or fewer calls in one hour? What is the probability that there are exactly 15 calls in two hours?

40 Another Poisson Problem  The number of automobile accidents on I-75 passing through downtown Dayton has a Poisson distribution with a mean of 3 per week. What is the probability of at least one accident a week?

41 Summary of Distributions DistributionMeanVarianceVariance / Mean Uniform (a+b) /2 [(b-a+1) 2 – 1] / 12 [(b-a+1) 2 – 1] / [2(b+a)] Binomialnpnp(1 - p)(1 – p) < 1 Geometric1/p(1 – p)/p 2 (1 – p) / p Neg. Binomial r/pr(1-p)/p 2 (1 - p) / p Hypergeom.np np(1-p)(N-n)/(N- 1) (1-p)(N-n)/(N-1) Poisson 1

42 Other Discrete Distributions Worth Knowing  Negative Binomial a generalization of the geometric distribution number of Bernoulli trials until the r th success  Hypergeometric used to model finite populations parameters are  N – population size  n – sample size  K – number of successes in population X = a discrete random variable, the number of “successes” in the sample probabilities computed from combinations

43 Next Week…Chapter 4