L06. Chapter 6: Continuous Probability Distributions

Transcription

1 L06 Chapter 6: Continuous Probability Distributions

2 Probability Chapter 6 Continuous Probability Distributions Recall Discrete Probability Distributions Could only take on particular values Continuous can take on any value Values of Random Variable (TV sales)

3 Continuous Probability Distributions Uniform Probability Distribution Normal Probability Distribution Eponential Probability Distribution f () Uniform f () Eponential f () Normal

4 Continuous Probability Distributions A continuous random variable can assume any value in an interval on the real line or in intervals. To find probabilities, we use areas under a probability density function It is not possible to talk about the probability of the random variable assuming a single value. For eample: Probability that height = 60 inches This is because the area under a single point is zero Instead, we talk about the probability of the random variable assuming a value within an For Eample, Height being between 60 and 65 inches

5 Continuous Probability Distributions The probability of the random variable assuming a value within an interval from 1 to 2 is defined to be the under the graph of the probability density function between 1 and 2. f () Uniform f () Eponential f () Normal

6 Uniform Probability Distribution A random variable is uniformly distributed whenever the probability is proportional to the interval s length. The uniform probability density function is: f () = 1/(b a) for a < < b = 0 elsewhere f () where: a = smallest value the variable can assume b = largest value the variable can assume These Statements tell us about the shape of the probability distribution To find Probabilities, we need the the shape

7 Uniform Probability Distribution Eample: Slater's Buffet Slater customers are charged for the amount of salad they take. Sampling suggests that the amount of salad taken is uniformly distributed between 5 ounces and 15 ounces.

8 Uniform Probability Distribution Uniform Probability Density Function f() = 1/10 for 5 < < 15 = 0 elsewhere where: = salad plate filling weight

9 Uniform Probability Distribution Uniform Probability Distribution for Salad Plate Filling Weight f() 1/ Salad Weight (oz.)

10 Uniform Probability Distribution What is the probability that a customer will take between 12 and 15 ounces of salad? f() 1/ Salad Weight (oz.) Notice, we simply used the formula for the area of a rectangle, BASE * HEIGHT

11 Uniform Probability Distribution Epected Value of E() = (a + b)/2 Variance of Var() = (b - a) 2 /12

12 Uniform Probability Distribution Epected Value of E() = (a + b)/2 = (5 + 15)/2 = 10 Variance of Var() = (b - a) 2 /12 = (15 5) 2 /12 = 8.33

13 Heights of people Test scores Normal Probability Distribution Is this chapter discrete or continuous? And how do we find the probability of variables that are continuous? We are staying in the world where we find probability by the area under a curve. We simply are of the curve Normal curve will be used etensively throughout the rest of this semester and net semester. Scientific measurements Amounts of rainfall

14 Normal Probability Distribution Let s take a look at what the curve looks like.

15 Normal Probability Function Let s take a look at the formula that generates our curve = mean = standard deviation = e =

16 Normal Probability Distribution Characteristics Distribution is Skew is Tails are of one another Value on the -ais below highest point is the mean, median, and mode.

17 Normal Probability Distribution Characteristics The mean can be any numerical value The mean moves the distribution to

18 Normal Probability Distribution Characteristics The standard deviation determines the of the curve. Greater standard deviation, the. = 15 = 25

19 Normal Probability Distribution Characteristics Probabilities = area under the curve. Total area = Area under right half = Same for left..5.5

20 Standard Normal Distribution There are many means for a normal distribution There are many standard deviations for the normal distribution. We are going to get our probability information from a table, but our book is not big enough to contain infinitely man normal distribution tables. What should we do? STANDARDIZE so we only have to use one table

21 Standard Normal Standard Normal Probability Distribution: A normal distribution with mean of 0 and standard deviation of 1 All normal distributions can be into the standard normal distribution We use the transformation so we don t have to have infinitely many tables in the back of the book. The letter z is used to designate the standard normal random variable

22 Standard Normal Transforming from normal to standard normal z Interpretation of z The number of is from the mean

23 Eample Now let s work on a problem where we have to go from a normal distribution to a standardized normal distribution. The time required to build a computer is normally distributed with a mean of 50 minutes and a standard deviation of 10 minutes. What is the probability that a computer is assembled in a time between 45 and 60 minutes? Algebraically speaking, what is P(45 < X < 60)? Method: 1. Draw 2. Convert to Z 3. Look up probabilities in Table 0

24 Eample CONVERT TO A STATEMENT ABOUT Z P(45<X<60) = P( < X < ) Draw in your z values Go to table Find area of interest. Answer = z = -.5 z = 1

25 Eponential Probability Distribution Useful to describe it takes to complete a task or for something to happen Time between vehicle arrivals at a toll booth Time required to complete a questionnaire Distance between major defects in a highway

26 Eponential Probability Distribution We are staying in the world where we find probability by the area under a curve. We simply are changing the shape of the curve Shape of the curve can be represented by the density function f ( ) 1 e / For 0, > 0 Think Plotting Points where: = mean e =

27 f ( ) 1 e / where: = mean e = / Total area under curve is

28 Eponential Distribution Variable is quantitative ( continuous). X values must be positive (or zero). Only one parameter: (mean) Std. deviation: = (same value) SKEWED right.

29 Eponential Probability Distribution How to work with Eponential Uniform we used the formula base * height to find the area Normal we used the table to find the area Eponential we use a to find the area P( ) 1 e o / This formula gives the area to the of 0 X 0 is some specific value of the variable 0

30 Relationship Between Poisson and Eponential Poisson Number of occurrences per interval Eponential TIME between occurrences Skill we want. Go from Poisson to Eponential Framework Number of cars that arrive at the carwash follows a Poisson Distribution with mean of 10 cars per hour. How do we go from cars per hour to hours per car? 10 cars/hour 1 hour / 10 cars =.1 hours per car This tells us for the eponential distribution Skill: When given poisson, we can to eponential by division.

31 Final Point on Eponential Remember in Ecel: Lambda = 1/ eponential