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Chapter 5: Continuous Random Variables

Chapter 5: Continuous Random Variables. Where We’ve Been. Using probability rules to find the probability of discrete events Examined probability models for discrete random variables. Where We’re Going. Develop the notion of a probability distribution for a continuous random variable

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Chapter 5: Continuous Random Variables

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  1. Chapter 5: Continuous Random Variables

  2. 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

  3. 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

  4. 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. 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

  6. 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

  7. 5.2: The Uniform Distribution Mean: Standard Deviation: McClave: Statistics, 11th ed. Chapter 5: Continuous Random Variables

  8. 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

  9. 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

  10. 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

  11. 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

  12. 5.3: The Normal Distribution McClave: Statistics, 11th ed. Chapter 5: Continuous Random Variables

  13. 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

  14. 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

  15. 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

  16. 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

  17. 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

  18. 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

  19. 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

  20. 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

  21. 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

  22. 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

  23. 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

  24. 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

  25. 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

  26. 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

  27. 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

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