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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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where we ve been
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
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
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 distributions1
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
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 distribution1
5.2: The Uniform Distribution

Mean:

Standard Deviation:

McClave: Statistics, 11th ed. Chapter 5: Continuous Random Variables

5 2 the uniform distribution2
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 distribution3
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
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 distribution1
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 distribution2
5.3: The Normal Distribution

McClave: Statistics, 11th ed. Chapter 5: Continuous Random Variables

5 3 the normal distribution3
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 distribution4
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 distribution5
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 distribution6
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
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 normality1
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 normality2
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
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

slide21

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 distribution1
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 distribution2
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 distribution3
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
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 distribution1
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 distribution2
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