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L06 Chapter 6: Continuous Probability Distributions Chapter 6 Continuous Probability Distributions Recall Discrete Probability Distributions ◦ Could only take on particular values ◦ Continuous can take on any value .50 Probability .40 .30 .20 .10 0 1 2 3 4 Values of Random Variable x (TV sales) Continuous Probability Distributions Uniform Probability Distribution Normal Probability Distribution Exponential Probability Distribution f (x) Uniform f (x) Exponential x f (x) Normal x x 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 example: 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 Example, Height being between 60 and 65 inches Continuous Probability Distributions The probability of the random variable assuming a value within an interval from x1 to x2 is defined to be the _____ under the graph of the probability density function between x1 and x2. f (x) f (x) Exponential Uniform f (x) x1 x2 Normal x1 xx12 x2 x x1 x2 x x 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 (x) = 1/(b – a) for a < x < b =0 elsewhere •where: f (x) 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 x Uniform Probability Distribution Example: 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. Uniform Probability Distribution Uniform Probability Density Function f(x) = 1/10 for 5 < x < 15 =0 elsewhere where: x = salad plate filling weight Uniform Probability Distribution Uniform Probability Distribution for Salad Plate Filling Weight f(x) 1/10 5 10 15 Salad Weight (oz.) x Uniform Probability Distribution What is the probability that a customer will take between 12 and 15 ounces of salad? f(x) 1/10 5 10 12 15 Salad Weight (oz.) x Notice, we simply used the formula for the area of a rectangle, BASE * HEIGHT Uniform Probability Distribution Expected Value of x E(x) = (a + b)/2 Variance of x Var(x) = (b - a)2/12 Uniform Probability Distribution Expected Value of x E(x) = (a + b)/2 = (5 + 15)/2 = 10 Variance of x Var(x) = (b - a)2/12 = (15 – 5)2/12 = 8.33 Heights of people Normal Probability Distribution Test scores Scientific measurements 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 Amounts under a curve. We simply of rainfall are ____________ ________ of the curve Normal curve will be used extensively throughout the rest of this semester and next semester. Normal Probability Distribution Let’s take a look at what the curve looks like. x Normal Probability Function Let’s take a look at the formula that generates our curve = mean = standard deviation = 3.14159 e = 2.71828 Normal Probability Distribution Characteristics Distribution is __________ ◦ Skew is _________ ◦ Tails are _____________of one another Value on the x-axis below highest point is the mean, median, and mode. x Normal Probability Distribution Characteristics ◦ The mean can be any numerical value ◦ The mean moves the distribution to _______________ x -10 0 20 Normal Probability Distribution Characteristics ◦ The standard deviation determines the ______of the curve. Greater standard deviation, _______ the _______. = 15 = 25 x Normal Probability Distribution Characteristics ◦ Probabilities = area under the curve. ◦ Total area =_______ ◦ Area under right half = _____Same for left. .5 .5 x 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 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 Standard Normal Transforming from “normal” to “standard normal” x z Interpretation of z ◦ The number of ________________x is from the mean Example 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 Example CONVERT TO A STATEMENT ABOUT Z P(45<X<60) = P( < Draw in your z values Go to table Find area of interest. ______________________________ Answer = ____________ X < ) z = -.5 z=1 Exponential 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 Exponential 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 ( x) 1 e x / For x ≥ 0, > 0 where: Think Plotting Points = mean e = 2.71828 f ( x) 1 e x / where: = mean 4 e = 2.71828 0.3000 1/ 0.2500 Total area under curve is 1.0 0.2000 0.1500 0.1000 0.0500 0.0000 0.0 4.0 8.0 12.0 16.0 20.0 24.0 28.0 Exponential Distribution • Variable is quantitative ( continuous). X values must be positive (or zero). • Only one parameter: • Std. deviation: • SK E = ED right. W (mean) (same value) Exponential Probability Distribution How to work with Exponential ◦ Uniform – we used the formula base * height to find the area ◦ Normal – we used the table to find the area ◦ Exponential – we use a _______ to find the area P( x xo ) 1 e x0 / ◦ This formula gives the area to the ______ of x0 ◦ X0 is some specific value of the variable x Relationship Between Poisson and Exponential Poisson Number of occurrences per interval Exponential TIME between occurrences Skill we want. Go from Poisson to Exponential 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 exponential distribution Skill: When given poisson, we can ______to exponential by division. Final Point on Exponential Remember in Excel: ◦ Lambda = 1/exponential