Chapter 5 Continuous Random Variables

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Chapter 5 Continuous Random Variables

• 5.1 Continuous Probability Distributions
• 5.2 The Uniform Distribution
• 5.3 The Normal Probability Distribution

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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).

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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.

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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 .

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Example 5.1

• Suppose we had a continuous random variable x
  with the following
• probability density function
• f(x) , 0 ? x ? 2
• and
• f(x) , 2 ? x ? 4
• If we wanted to determine the probability that x
  falls between 3 and 4.

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Step 1. Draw the function f(x).
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Step 2 Identify the Interval of Interest
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Step 3 Shade in the area under f(x) in this
interval
.125
1 unit
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Step 4 Compute the area of the shaded region.

• The shaded region is a triangle. The base equals
  1. The height equals
• 0.125 (You can verify this by plugging x 3 into
  the function f(x)).
• Area 1/2 bh ½ 1 0.125 0.0625
• Thus the probability that x falls between 3 and 4
  equals 0.0625.

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Special Random Variables

• In this chapter we will discuss two popular
  continuous random variables
• Uniform random variable
• Normal random variable

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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/(d - c) for c lt x lt d
• 0 elsewhere
• where c smallest value the variable can
  assume
• d largest value the variable can
  assume

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Uniform Probability Distribution

• Expected Value of x
• E(x) (c d)/2
• Variance of x
• Var(x) (d - c)2/12

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Example 5.2

• 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

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What is the probability that the time that it
will take to service a customer is between 12 to
15 minutes ?
15
Width 3
Height0.1
16

• 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.

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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
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Example 5.2 (Expected Value and Variance)

• Expected Service time
• (5 15)/2 10 minutes
• Variance of Service times
• (15-5)2 /12 8.33 minutes

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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.

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Normal Probability Distribution

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

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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.

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

Bowerman, et. al
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Probabilities for the normal random variable are
given by areas under the curve.
Bowerman, et. al
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The Position and Shape of the Normal Curve
Bowerman, et. al
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• 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
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• In order to better understand the normal
• probability distribution you should STOP
• here, go to Course Documents, click on
• Chapter 5, and complete the Normal
• Distribution Exercise. Check your answers,
• and then return to Chapter 5 part 2
• notes.