Continuous probability distribution

Transcription

1 Microarray Center BIOSTATISTICS Lecture 6 Continuous Probability Distributions Lecture 6. Continuous probability distributions Dr. Petr Nazarov petr.nazarov@crp-sante.lu

2 OUTLINE Lecture 1 Continuous probability distribution a continuous probability distribution uniform probability distribution normal probability distribution eponential probability distribution Lecture 6. Continuous probability distributions

3 RANDOM VARIABLES Random Variables Random Random variable variable A numerical numerical description description of of the the outcome outcome of of an an eperiment. eperiment. A random variable is always a numerical measure. Roll a die Number of calls to a reception per hour Discrete Discrete random random variable variable A random random variable variable that that may may assume assume either either a finite finite number number of of values values or or an an infinite infinite sequence sequence of of values. values. Continuous Continuous random random variable variable A random random variable variable that that may may assume assume any any numerical numerical value value in in an an interval interval or or collection collection of of intervals. intervals. Time between calls to a reception Volume of a sample in a tube Weight, height, blood pressure, etc Lecture 6. Continuous probability distributions 3

4 Probability Density Probability Probability density density function function A function function used used to to compute compute probabilities probabilities for for a continuous continuous random randomvariable. The The area area under under the the graph graph of of a probability probability density density function function over over an an interval interval represents represents probability. probability..3.3 Probability Probability density density Area 1 f ( ) Variable Variable Lecture 6. Continuous probability distributions 4

5 Uniform Probability Distribution Uniform Uniform probability probability distribution distribution A continuous continuous probability probability distribution distribution for for which which the the probability probability that that the the random random variable variable will will assume assume a value value in in any any interval interval is is the the same same for for each each interval interval of of equal equal length. length. f ( ) b 1 a,, for a elsewhere b E( ) a + b µ ( b a) Var( ) σ 1 Eample The bus goes every 7 minutes. You are coming to CHL bus station, having no idea about precise timetable. What is the distribution for the time, you may wait there? Lecture 6. Continuous probability distributions 5

6 Normal Probability Distribution Normal Normal probability probability distribution distribution A continuous continuous probability probability distribution. distribution. Its Its probability probability density density function function is is bell bell shaped shaped and and determined determined by by its its mean mean µ µ and and standard standard deviation deviation σ. σ. f ( µ ) 1 σ ( ) e σ π In Ecel use the function: NORMDIST(,m,s,false) for probability density function NORMDIST(,m,s,true) for cumulative probability function of normal distribution (area from left to ) Lecture 6. Continuous probability distributions 6

7 Standard Normal Probability Distribution Standard Standard normal normal probability probability distribution distribution A normal normal distribution distribution with with a mean mean of of zero zero and and a standard standard deviation deviation of of one. one. f ( ) 1 e π z µ σ In Ecel use the function: NORMSDIST(z)-.5 Lecture 6. Continuous probability distributions 7

8 Eample: Gear Tire Company Eample Eample Suppose Suppose the the Grear GrearTire Tire Company Company just just developed developed a new new steel-belted steel-belted radial radial tire tire that that will will be be sold sold through through a chain chain of of discount discount stores. stores. Because Because the the tire tire is is a new new product, product, Grear's Grear'smanagers believe believe that that the the mileage mileage guarantee guarantee offered offered with with the the tire tire will will be bean an important important factor factor in in the the acceptance acceptance of of the the product. product. Before Before finalizing finalizing the the tire tire mileage mileage guarantee guarantee policy, policy, Grear's Grear's managers managers want want probability probability information information about about the the number number of of miles miles the the tires tires will will last. last. From From actual actual road road tests tests with with the the tires, tires, Grear's Grear'sengineering engineering group group estimates estimates the the mean mean tire tire mileage mileage is is µ miles miles with with a standard standard deviation deviation of of σ 5.. In In addition, addition, data data collected collected indicate indicate a normal normal distribution distribution is is a reasonable reasonable assumption. assumption. What What percentage percentage of of the the tires tires can can be be epected epected to to last last more more than than 4 4 miles? miles? In In other other words, words, what what is is the the probability probability that that a tire tire mileage mileage will will eceed eceed 4 4?? Anderson et al Statistics for Business and Economics Lecture 6. Continuous probability distributions 8

9 Eample: Gear Tire Company 1. Let s transfer from Normal distribution to Standard Normal, then z, corresponding to 4 will be z Calculate the blue area P(z >.7) using the table: P(z>.7) 1 P(z<.7) 1.5 P(<z<.7) Alternatively in Ecel 1-NORMDIST(4,365,5,true) Lecture 6. Continuous probability distributions 9

10 Eponential Eponential probability probability distribution distribution A continuous continuous probability probability distribution distribution that that is is useful useful in in computing computing probabilities probabilities for for the the time time between between independent independent random random events. events. CONTINUOUS PROBABILITY DISTRIBUTIONS Eponential Probability Distribution Eample Number of calls to an Emergency Service is on average 3 per hour b/w. and 6. of working days. What are the distribution of the time between the calls? Time between calls to a reception µ 1 σ λ f 1 µ ( ) e for, µ > µ f ( ) λe λ Cumulative probability function P( ) F( ) 1 e µ Lecture 6. Continuous probability distributions 1

11 Eample: Eponential Distribution for Fish Counting Eample Eample An An ichthyologist ichthyologist studying studying the the spoonhead spoonheadsculpin sculpincatches catches specimens specimens in in a large large bag bag seine seine that that she she trolls trolls through through the the lake. lake. She She knows knows from from many many years years eperience eperience that that on on averages averages she she will will catch catch fish fish per per trolling. trolling. Each Each trolling trolling take take ~3 ~3 minutes. minutes. Find Find the the probability probability of of catching catching no no fish fish in in the the net net hour hour In Ecel use the function: EXPONDIST(,1/mu,false) 1. Let s calculate µ for this situation: µ 3 / 15 minutes P.D.F. P.D.F. Eponential distribution with mu15 Eponential distribution with mu Time between getting a fish, min Time between getting a fish, min. Use either a cumulative probability function or Ecel to calculate: 6 15 P( 6) 1 P( 6) 1 F(6) e. Lecture 6. Continuous probability distributions 11

12 QUESTIONS? Thank you for your attention to be continued Lecture 6. Continuous probability distributions 1