1 Ch. 3.4-3.5 IOE/Stat 265, Fall 2009 Lecture #7: Binomial Distribution (and Relatives) 2 Common Discrete Distributions Discrete Uniform 3.4 Binomial Special Case: Bernoulli Geometric Negative Geometric 3.5 Hypergeometric & Negative Binomial 3.6 Poisson 3 Discrete Uniform Distribution � A random variable X has a discrete uniform distribution if p(x) = 1/n for x 1 , x 2 , …, x n () ⎧ + ⎪ = ⎨ ⎪ ⎩ + μ= = − +− σ= −μ = K 2 22 1 a,a 1, ,b p(x) n 0 otherwise ba E(x) 2 ba1 1 E(x ) 12 0246810 X 0 0.05 0.1 0.15 0.2 Discrete Uniform Probability Mass Function a=1 b=10 4 Example 1: Search & Rescue � It’s 40 miles to DTW airport. Your car has broken down on the way to catch a plane. You call a tow truck for assistance and describe your location based on a nearby mile marker. � What is the probability that you are less than 10 miles from the airport? � What is the standard deviation of distance? 7 3-4 Bernoulli Distribution � Most fundamental distribution: The number of successes in one Bernoulli trial. We assume the probability of success, p, is known and that the trial can result in only one of two mutually exclusive outcomes – success or failure. � Bernoulli trial? A random experiment with only two possible outcomes, "success" and "failure" with probabilities p and (1-p). � p(x) = p x (1 – p) 1-x x = 0, 1 0 ≤ p ≤ 1 � μ = E(x) = p σ 2 = E(x- μ) 2 = p (1–p) 8 p = 0.1 p = 0.2 p = 0.5 0 0.2 0.4 0.6 0.8 1 X 0 0.2 0.4 0.6 0.8 1 Bernoulli Probability Mass Function 9 Example 2: Defective Parts � Suppose 0.1% of production parts are defective. What is the probability that the first item inspected will be nondefective? 11 3-4 Binomial Distribution � A binomial random variable arises with “n” repeated trials of a Bernoulli experiment. � Characteristics: � n trials, fixed in advance � Identical trials, with only 2 possible outcomes (Bernoulli trial) � Independent trials � Probability of success (p) is constant 12 Examples � Flipping a coin 10 times � Testing a cell phone network for transmission or connection errors � Testing a sample of air for a particular pollutant � Number of female births at a hospital 13 Probability Mass Function � The PMF of a binomial random variable is denoted in terms of parameters: � n (number of trials) and � p (probability of success) () () − ⎧ ⎛⎞ −= ⎪ ⎜⎟ = ⎨ ⎝⎠ ⎪ ⎩ K nx x n p1p x 0,1,2, ,n px x 0 otherwise 14 Mean and Variance � If X ~ Bin(n,p) then the mean and the variance of X are defined as: () () 2 EX np np 1 p npq where q 1 p npq μ= = σ= − = =− σ= 15 Shape of Binomial Distribution � Generate a set of binomial random numbers and plot in Minitab 02 46810 X 0 0.1 0.2 0.3 0.4 Binomial Probability Mass Function n = 10 p = 0.1 n = 10 p = 0.2 n = 10 p = 0.3 n = 10 p = 0.5 0 1020304050 X 0 0.04 0.08 0.12 0.16 0.2 Probability Mass Binomial Probability Mass Function n = 50 p = 0.1 n = 50 p = 0.2 n = 50 p = 0.3 n = 50 p = 0.5 16 Binomial Tables � Appendix Table A.1 in textbook tabulates the binomial cdf P(X ≤ x) for n=1…15, 20, 25 with selected values of p () () x ny y y0 n P X x p 1 p x 0,1, ,n y − = ⎛⎞ ≤= − = ⎜⎟ ⎝⎠ ∑ K 17 Excel Functions � =BINOMDIST ( x , n , p , CDF ) � If CDF is TRUE the function returns the cumulative distribution function CDF. � If CDF is FALSE the function returns the probability mass function, PMF. � =CRITBINOM ( n , p , α) � Returns the smallest value, x, for which the binomial cumulative distribution is less than or equal to a criterion value, α. 18 Example 3: Air Pollution � Air pollution monitoring in a big city is underway. Suppose there is a 10% chance of finding a high concentration. You take 18 samples at random. � What is: � Pr(exactly 2 samples have high conc) = ? � Pr(at least 4)=? � Pr(at least 3 but at most 6)=? 22 Example 4: Toll Bridge Revenue � A toll bridge charges $1.00 for passenger cars and $2.50 for other vehicles. Suppose that during day time 60% of all vehicles are passenger cars. If 25 vehicles cross during a particular daytime, what is the expected toll revenue? 24 Example 4: EXTRA CREDIT � At the end of the day there is less than $30 in the register… would you suspect someone of sticky fingers? 26 Example 5: Quality Assurance � A firm claims that 99% of their products meet specifications. To support this claim, an inspector draws a random sample of 20 items and ships the lot if the entire sample is in conformance. Find the probability of committing each of the following errors: � Refusing to ship a lot even though 99% of the items are in conformance. � Shipping a lot even though only 95% of the items are conforming. 30 3-5 Geometric Distribution � A random variable X follows a geometric distribution if X denotes the number of independent Bernoulli trials until the first success. The probability of success, p, is constant. The PMF is: ( ) ( ) () () () − =− = μ= = − σ= −μ = K x1 2 2 2 p x 1 p p x 1,2, ,n 1 EX p 1p EX p p = 0.1 p = 0.3 p = 0.5 p = 0.8 0246810 X 0 0.2 0.4 0.6 0.8 Geometric Probability Mass Function 31 Example 6: Pipeline Inspection � Suppose 2% of seams on a pipeline are showing signs of wear. On the average, how many seams will be inspected before finding the first worn seam? 33 Example Continued � Seams are 50 feet apart along the pipeline. � What is the probability the inspector will walk further than 5000 feet before finding a worn seam. 35 3-5 Negative Binomial Distribution � Also known as “Pascal Distribution” � A Negative Binomial rv satisfies the following: � Independent trials � Bernoulli trials – success (S) or failure (F) � Probability of success is constant (p) � The experiment continues until a total of r successes have been observed, r is a specified positive integer 36 Why called “Negative Binomial”? � With Binomial “n” is constant (given) and “X” is variable we solve for. � With Negative Binomial “X” is constant (given) and “n” is variable. 37 Two Forms � Form 1: X = Number of trials until the r th success. � Form 2: X = Number of failures prior to the r th success. � EXCEL: =NEGBINOMDIST(x, r, p) � assumes Form 2 38 Negative Binomial Distribution (Form 1) � This form not used by Textbook (see p 119) � p= probability of success. � X = number of trials to r th success. r = 1, 2, …. � The PMF of the negative binomial with parameters r and p is: () ( ) () () () −−⎛⎞ =−=+ ⎜⎟ − ⎝⎠ = − = K xr r 2 x1 px p 1 p x r,r 1, r1 r EX p r1 p VX p 39 Negative Binomial Distribution (Form 2) � Used by Textbook � X = number of failures prior to rth success. � The PMF of the negative binomial with parameters r and p is: () ( ) () () () () +−⎛⎞ =−= ⎜⎟ − ⎝⎠ − = − = K x r 2 xr1 px p 1 p x 0,1,2, r1 r1 p EX p r1 p VX p 40 � Note: If r = 1, then we have the “Geometric Distribution" (the number of Bernoulli trials until the first success). r = 10, p = 0.1 r = 10, p = 0.3 r = 10, p = 0.5 0 30 60 90 120 150 X 0 0.02 0.04 0.06 0.08 0.1 Probabil i t y Mass Function Negative Binomial Probability Mass Function 41 Example 7: Family Planning � A family wants to have children until they have two female children. If Pr(male birth)=0.5, � P(family has 4 children)=? � P(family has at most 4 children)=? � How many male children should they expect? 0.25 0.20 0.15 0.10 0.05 0.00 X P r o b a b ilit y 4 0.688 13 Distribution Plot Negative Binomial, p=0.5, NEvents=2 X = total number of trials. 47 Example 8: Web Servers � Web service has 3 servers (1 primary, 2 backups). Prob of failure=0.0005 for each. � What is the expected time to failure? � What probability of failure within 5 requests? 50 p 0.0005 r3 x = 0 1.250E-10 =NEGBINOMDIST(B4,$B$2,$B$1) 1 3.748E-10 =NEGBINOMDIST(B5,$B$2,$B$1) 2 7.493E-10 =NEGBINOMDIST(B6,$B$2,$B$1) 1.249E-09 =SUM(D4:D6) 51 Summary: Uniform, Binomial, Geometric or Negative Binomial? � Uniform → all X outcomes are equilikely � Binomial → n independent Bernoulli (S/F) trials with fixed p. � Bernoulli Distribution → Binomial with n=1 � Geometric → independent Bernoulli trials with fixed p until the first success. � Negative Binomial → independent Bernoulli trials with fixed p until r successes. � Geometric is special case with r=1 gdherrin Microsoft PowerPoint - Lec07 - Binomial