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Chapter 6 Continuous Random Variables

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Title: Chapter 6 Continuous Random Variables


1
Chapter 6 Continuous Random Variables
  • Continuous Probability Distributions
  • The Uniform Distribution
  • The Normal Probability Distribution

2
Continuous Probability Distributions
  • A continuous random variable can assume any value
    in an interval on the real line or in a
    collection of intervals.
  • The function f(x) is the probability density
    function (or probability distribution function)
    of the continuous random variable x.
  • Unlike a discrete random variable, we can not
    simply plug values of the random variable into
    this function and get probability information
    directly.
  • For continuous random variables it is impossible
    to talk about the probability of the random
    variable assuming a particular value.
  • Instead, we talk about the probability of the
    random variable assuming a value within a given
    interval.
  • In order to determine the probability that a
    continuous random variable assumes a value in an
    interval you must first draw the function f(x).

3
Properties of a Continuous Random Variable
  • The probability density function f(x)?0 for all
    values of x.
  • The probability of a continuous random variable
    assuming a value within some given interval from
    x1 to x2 is defined to be the area under the
    graph of the probability density function between
    x1 and x2.
  • The probability of a continuous random variable
    assuming a specific value is zero (there is no
    area under any graph at an exact point).
  • The total area under the graph of f(x) equals 1.

4
How to find the Probability that a Continuous
Random Variable Falls within an Interval?
  • In order to find the probability that a
    continuous random variable, x falls
  • in an interval between x1 and x2 do the
    following
  • Graph the probability density function.
  • Identify the interval of interest on the x axis.
  • Shade in the area under f(x) in this interval.
  • Compute the area of the shaded region.
  • The area of the shaded region is the probability
    that x will fall between x1 and x2 .
  • The probability that x falls in the interval x1
    to x2 is the same as the
  • proportion of x values from the population that
    fall between x1 and x2 .

5
Special Random Variables
  • In this chapter we will discuss two popular
    continuous random variables
  • Uniform random variable
  • Normal random variable

6
Uniform Probability Distribution
  • A random variable is uniformly distributed
    whenever it is equally likely that a random
    variable could take on any value between c and d.
  • The uniform random variable has the following
    probability density function
  • f(x) 1/(b - a) for a lt x lt b
  • 0 elsewhere
  • where a smallest value the variable can
    assume
  • b largest value the variable can
    assume

7
Uniform Probability Distribution
  • Expected Value of x
  • E(x) (a b)/2
  • Variance of x
  • Var(x) (b - a)2/12

8
Example 6.2
  • The service time at a CALs restaurant is
    uniformly distributed between 5
  • and 15 minutes.
  • The probability density function is
  • f(x) 1/10 for 5 lt x lt 15
  • 0 elsewhere
  • where
  • x the service time for a customer

9
What is the probability that the time that it
will take to service a customer is between 12 to
15 minutes ?
10
Width 3
Height0.1
11
  • The area under f(x) between 12 and 15 is a
    rectangle.
  • area 3 0.10.3
  • Thus the probability that it takes between 12 to
    15 minutes for a customer
  • to get serviced is 0.3. Moreover, we can
    conclude that 30 of the CALs
  • customers will wait between 12 to 15 minutes for
    service.

12
What is the probability that the time that it
will take to service a customer is between 7 to
12 minutes ?
5 0.1 0.5
13
Example 6.2 (Expected Value and Variance)
  • Expected Service time
  • (5 15)/2 10 minutes
  • Variance of Service times
  • (15-5)2 /12 8.33 minutes

14
Normal Probability Distribution
  • The normal probability distribution is the most
    popular and important distribution for describing
    a continuous random variable.
  • This distribution has been used to define
  • This distribution is used in various statistical
    inference techniques
  • Heights and weights of people
  • Test scores
  • IQ Scores
  • The Normal distribution is widely used in various
    statistical inference techniques.

15
Normal Probability Distribution
  • The probability density function for a normal
    random variable is
  • where
  • ? mean
  • ? standard deviation
  • ? 3.14159
  • e 2.71828

16
Characteristics of the Normal Probability
Distribution
  • The distribution is symmetric, and illustrated
    as a bell-shaped curve.
  • Two parameters, ? (mean) and ? (standard
    deviation), determine the location and shape of
    the distribution.
  • The highest point on the normal curve is at the
    mean, which is also the median and mode.
  • The mean can be any numerical value negative,
    zero, or positive.

17
  • The total area under the curve is 1 (.5 to the
    left of the mean and .5 to the right).

Bowerman, et. al
18
Probabilities for the normal random variable are
given by areas under the curve.
Bowerman, et. al
19
The Position and Shape of the Normal Curve
Bowerman, et. al
20
  • 68.26 of values of a normal random variable are
    within /- 1 standard deviation of its mean.
  • 95.44 of values of a normal random variable are
    within /- 2 standard deviations of its mean.
  • 99.73 of values of a normal random variable are
    within /- 3 standard deviations of its mean.

Bowerman, et. al
21
  • In order to better understand the normal
  • probability distribution you should STOP
  • here, go to Course Documents, click on
  • Chapter 6, and complete the Normal
  • Distribution Exercise. Check your answers,
  • and then return to Chapter 6 part 2
  • notes.
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