Chapter 5: Continuous Random Variables
This chapter delves into continuous random variables, exploring their probability distributions and the significance of the normal distribution in statistics. Key concepts include the definition of continuous probability distributions, the uniform distribution, and the properties of the normal distribution. The chapter explains how to identify and analyze continuous variables through probability density functions, highlight the role of integral calculus in finding probabilities, and introduce descriptive methods for assessing normality in data analysis.
Chapter 5: Continuous Random Variables
E N D
Presentation Transcript
Where We’ve Been • Using probability rules to find the probability of discrete events • Examined probability models for discrete random variables McClave: Statistics, 11th ed. Chapter 5: Continuous Random Variables
Where We’re Going • Develop the notion of a probability distribution for a continuous random variable • Examine several important continuous random variables and their probability models • Introduce the normal probability distribution McClave: Statistics, 11th ed. Chapter 5: Continuous Random Variables
5.1: Continuous Probability Distributions • A continuousrandom variable can assume any numerical value within some interval or intervals. • The graph of the probability distribution is a smooth curve called a • probability density function, • frequency function or • probability distribution. McClave: Statistics, 11th ed. Chapter 5: Continuous Random Variables
5.1: Continuous Probability Distributions • There are an infinite number of possible outcomes • p(x) = 0 • Instead, find p(a<x<b) Table Software Integral calculus) McClave: Statistics, 11th ed. Chapter 5: Continuous Random Variables
5.2: The Uniform Distribution • X can take on any value between c and d with equal probability = 1/(d - c) • For two values a and b McClave: Statistics, 11th ed. Chapter 5: Continuous Random Variables
5.2: The Uniform Distribution Mean: Standard Deviation: McClave: Statistics, 11th ed. Chapter 5: Continuous Random Variables
5.2: The Uniform Distribution Suppose a random variable x is distributed uniformly with c = 5 and d = 25. What is P(10 x 18)? McClave: Statistics, 11th ed. Chapter 5: Continuous Random Variables
5.2: The Uniform Distribution Suppose a random variable x is distributed uniformly with c = 5 and d = 25. What is P(10 x 18)? McClave: Statistics, 11th ed. Chapter 5: Continuous Random Variables
5.3: The Normal Distribution • The probability density function f(x): µ = the mean of x = the standard deviation of x = 3.1416… e = 2.71828 … • Closely approximates many situations • Perfectly symmetrical around its mean McClave: Statistics, 11th ed. Chapter 5: Continuous Random Variables
5.3: The Normal Distribution • Each combination of µ and produces a unique normal curve • The standard normal curve is used in practice, based on the standard normal random variable z (µ = 0, = 1), with the probability distribution The probabilities for z are given in Table IV McClave: Statistics, 11th ed. Chapter 5: Continuous Random Variables
5.3: The Normal Distribution McClave: Statistics, 11th ed. Chapter 5: Continuous Random Variables
5.3: The Normal Distribution For a normally distributed random variable x, if we know µ and , So any normally distributed variable can be analyzed with this single distribution McClave: Statistics, 11th ed. Chapter 5: Continuous Random Variables
5.3: The Normal Distribution • Say a toy car goes an average of 3,000 yards between recharges, with a standard deviation of 50 yards (i.e., µ = 3,000 and = 50) • What is the probability that the car will go more than 3,100 yards without recharging? McClave: Statistics, 11th ed. Chapter 5: Continuous Random Variables
5.3: The Normal Distribution • Say a toy car goes an average of 3,000 yards between recharges, with a standard deviation of 50 yards (i.e., µ = 3,000 and = 50) • What is the probability that the car will go more than 3,100 yards without recharging? McClave: Statistics, 11th ed. Chapter 5: Continuous Random Variables
5.3: The Normal Distribution • To find the probability for a normal random variable … • Sketch the normal distribution • Indicate x’s mean • Convert the x variables into z values • Put both sets of values on the sketch, z below x • Use Table IV to find the desired probabilities McClave: Statistics, 11th ed. Chapter 5: Continuous Random Variables
5.4: Descriptive Methods for Assessing Normality • If the data are normal • A histogram or stem-and-leaf display will look like the normal curve • The mean ± s, 2s and 3s will approximate the empirical rule percentages • The ratio of the interquartile range to the standard deviation will be about 1.3 • A normal probability plot , a scatterplot with the ranked data on one axis and the expected z-scores from a standard normal distribution on the other axis, will produce close to a straight line McClave: Statistics, 11th ed. Chapter 5: Continuous Random Variables
5.4: Descriptive Methods for Assessing Normality Errors per MLB team in 2003 • Mean: 106 • Standard Deviation: 17 • IQR: 22 22 out of 30: 73% 28 out of 30: 93% 30 out of 30: 100% McClave: Statistics, 11th ed. Chapter 5: Continuous Random Variables
5.4: Descriptive Methods for Assessing Normality A normal probability plot is a scatterplot with the ranked data on one axis and the expected z-scores from a standard normal distribution on the other axis McClave: Statistics, 11th ed. Chapter 5: Continuous Random Variables
5.5: Approximating a Binomial Distribution with the Normal Distribution • Discrete calculations may become very cumbersome • The normal distribution may be used to approximate discrete distributions • The larger n is, and the closer p is to .5, the better the approximation • Since we need a range, not a value, the correction for continuity must be used • A number r becomes r+.5 McClave: Statistics, 11th ed. Chapter 5: Continuous Random Variables
5.5: Approximating a Binomial Distribution with the Normal Distribution Calculate the mean plus/minus 3 standard deviations If this interval is in the range 0 to n, the approximation will be reasonably close Express the binomial probability as a range of values Find the z-values for each binomial value Use the standard normal distribution to find the probability for the range of values you calculated McClave: Statistics, 11th ed. Chapter 5: Continuous Random Variables
5.5: Approximating a Binomial Distribution with the Normal Distribution Flip a coin 100 times and compare the binomial and normal results Binomial:Normal: McClave: Statistics, 11th ed. Chapter 5: Continuous Random Variables
5.5: Approximating a Binomial Distribution with the Normal Distribution Flip a weighted coin [P(H)=.4] 10 times and compare the results Binomial:Normal: McClave: Statistics, 11th ed. Chapter 5: Continuous Random Variables
5.5: Approximating a Binomial Distribution with the Normal Distribution Flip a weighted coin [P(H)=.4] 10 times and compare the results Binomial:Normal: The more p differs from .5, and the smaller n is, the less precise the approximation will be McClave: Statistics, 11th ed. Chapter 5: Continuous Random Variables
5.6: The Exponential Distribution • Probability Distribution for an Exponential Random Variable x • Probability Density Function • Mean: µ = • Standard Deviation: = McClave: Statistics, 11th ed. Chapter 5: Continuous Random Variables
5.6: The Exponential Distribution Suppose the waiting time to see the nurse at the student health center is distributed exponentially with a mean of 45 minutes. What is the probability that a student will wait more than an hour to get his or her generic pill? 60 McClave: Statistics, 11th ed. Chapter 5: Continuous Random Variables
5.6: The Exponential Distribution Suppose the waiting time to see the nurse at the student health center is distributed exponentially with a mean of 45 minutes. What is the probability that a student will wait more than an hour to get his or her generic pill? 60 McClave: Statistics, 11th ed. Chapter 5: Continuous Random Variables
