Continuous Probability Distributions Basics
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Oct 10, 2025
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About This Presentation
Continuous Probability Distributions Basics with examples
Slide Content
4-1 Continuous Random Variables
4-2 Probability Distributions and
Probability Density Functions
Figure 4-1 Density function of a loading on a long,
thin beam.
Probability Density Functions
Figure 4-1 Density function of a loading on a long,
thin beam.
4-2 Probability Distributions and
Probability Density Functions
Figure 4-2 Probability determined from the area
under f(x).
Probability Density Functions
Figure 4-2 Probability determined from the area
under f(x).
4-2 Probability Distributions and
Probability Density Functions
Definition
Probability Density Functions
Definition
4-2 Probability Distributions and
Probability Density Functions
Figure 4-3 Histogram approximates a probability
density function.
Probability Density Functions
Figure 4-3 Histogram approximates a probability
density function.
4-2 Probability Distributions and
Probability Density Functions
Probability Density Functions
4-2 Probability Distributions and
Probability Density Functions
Example 4-2
Probability Density Functions
Example 4-2
4-2 Probability Distributions and
Probability Density Functions
Figure 4-5 Probability density function for Example 4-2.
Probability Density Functions
Figure 4-5 Probability density function for Example 4-2.
4-2 Probability Distributions and
Probability Density Functions
Example 4-2 (continued)
Probability Density Functions
Example 4-2 (continued)
4-3 Cumulative Distribution Functions
Definition
Definition
4-3 Cumulative Distribution Functions
Example 4-4
Example 4-4
4-3 Cumulative Distribution Functions
Figure 4-7 Cumulative distribution function for Example
4-4.
Figure 4-7 Cumulative distribution function for Example
4-4.
4-4 Mean and Variance of a Continuous
Random Variable
Definition
Random Variable
Definition
4-4 Mean and Variance of a Continuous
Random Variable
Example 4-6
Random Variable
Example 4-6
4-4 Mean and Variance of a Continuous
Random Variable
Expected Value of a Function of a Continuous
Random Variable
Random Variable
Expected Value of a Function of a Continuous
Random Variable
4-4 Mean and Variance of a Continuous
Random Variable
Example 4-8
Random Variable
Example 4-8
4-5 Continuous Uniform Random
Variable
Definition
Variable
Definition
4-5 Continuous Uniform Random
Variable
Figure 4-8 Continuous uniform probability density
function.
Variable
Figure 4-8 Continuous uniform probability density
function.
4-5 Continuous Uniform Random
Variable
Mean and Variance
Variable
Mean and Variance
4-5 Continuous Uniform Random
Variable
Example 4-9
Variable
Example 4-9
4-5 Continuous Uniform Random
Variable
Figure 4-9 Probability for Example 4-9.
Variable
Figure 4-9 Probability for Example 4-9.
4-5 Continuous Uniform Random
Variable
Variable
4-6 Normal Distribution
Definition
Definition
4-6 Normal Distribution
Figure 4-10 Normal probability density functions for
selected values of the parameters and
2
.
Figure 4-10 Normal probability density functions for
selected values of the parameters and
2
.
4-6 Normal Distribution
Some useful results concerning the normal distribution
Some useful results concerning the normal distribution
4-6 Normal Distribution
Definition : Standard Normal
Definition : Standard Normal
4-6 Normal Distribution
Example 4-11
Figure 4-13 Standard normal probability density function.
Example 4-11
Figure 4-13 Standard normal probability density function.
4-6 Normal Distribution
Standardizing
Standardizing
4-6 Normal Distribution
Example 4-13
Example 4-13
4-6 Normal Distribution
Figure 4-15 Standardizing a normal random variable.
Figure 4-15 Standardizing a normal random variable.
4-6 Normal Distribution
To Calculate Probability
To Calculate Probability
4-6 Normal Distribution
Example 4-14
Example 4-14
4-6 Normal Distribution
Example 4-14 (continued)
Example 4-14 (continued)
4-6 Normal Distribution
Example 4-14 (continued)
Figure 4-16 Determining the value of x to meet a specified
probability.
Example 4-14 (continued)
Figure 4-16 Determining the value of x to meet a specified
probability.
4-7 Normal Approximation to the
Binomial and Poisson Distributions
• Under certain conditions, the normal
distribution can be used to approximate the
binomial distribution and the Poisson
distribution.
Binomial and Poisson Distributions
• Under certain conditions, the normal
distribution can be used to approximate the
binomial distribution and the Poisson
distribution.
4-7 Normal Approximation to the
Binomial and Poisson Distributions
Figure 4-19 Normal
approximation to the
binomial.
Binomial and Poisson Distributions
Figure 4-19 Normal
approximation to the
binomial.
4-7 Normal Approximation to the
Binomial and Poisson Distributions
Example 4-17
Binomial and Poisson Distributions
Example 4-17
4-7 Normal Approximation to the
Binomial and Poisson Distributions
Normal Approximation to the Binomial Distribution
Binomial and Poisson Distributions
Normal Approximation to the Binomial Distribution
4-7 Normal Approximation to the
Binomial and Poisson Distributions
Example 4-18
Binomial and Poisson Distributions
Example 4-18
4-7 Normal Approximation to the
Binomial and Poisson Distributions
Figure 4-21 Conditions for approximating hypergeometric
and binomial probabilities.
Binomial and Poisson Distributions
Figure 4-21 Conditions for approximating hypergeometric
and binomial probabilities.
4-7 Normal Approximation to the
Binomial and Poisson Distributions
Normal Approximation to the Poisson Distribution
Binomial and Poisson Distributions
Normal Approximation to the Poisson Distribution
4-7 Normal Approximation to the
Binomial and Poisson Distributions
Example 4-20
Binomial and Poisson Distributions
Example 4-20
4-8 Exponential Distribution
Definition
Definition
4-8 Exponential Distribution
Mean and Variance
Mean and Variance
4-8 Exponential Distribution
Example 4-21
Example 4-21
4-8 Exponential Distribution
Figure 4-23 Probability for the exponential distribution in
Example 4-21.
Figure 4-23 Probability for the exponential distribution in
Example 4-21.
4-8 Exponential Distribution
Example 4-21 (continued)
Example 4-21 (continued)
4-8 Exponential Distribution
Example 4-21 (continued)
Example 4-21 (continued)
4-8 Exponential Distribution
Example 4-21 (continued)
Example 4-21 (continued)
4-8 Exponential Distribution
Our starting point for observing the system does not matter.
•An even more interesting property of an exponential random
variable is the lack of memory property.
In Example 4-21, suppose that there are no log-ons
from 12:00 to 12:15; the probability that there are no
log-ons from 12:15 to 12:21 is still 0.082. Because we
have already been waiting for 15 minutes, we feel
that we are “due.” That is, the probability of a log-on
in the next 6 minutes should be greater than 0.082.
However, for an exponential distribution this is not
true.
Our starting point for observing the system does not matter.
•An even more interesting property of an exponential random
variable is the lack of memory property.
In Example 4-21, suppose that there are no log-ons
from 12:00 to 12:15; the probability that there are no
log-ons from 12:15 to 12:21 is still 0.082. Because we
have already been waiting for 15 minutes, we feel
that we are “due.” That is, the probability of a log-on
in the next 6 minutes should be greater than 0.082.
However, for an exponential distribution this is not
true.
4-8 Exponential Distribution
Example 4-22
Example 4-22
4-8 Exponential Distribution
Example 4-22 (continued)
Example 4-22 (continued)
4-8 Exponential Distribution
Example 4-22 (continued)
Example 4-22 (continued)
4-8 Exponential Distribution
Lack of Memory Property
Lack of Memory Property
4-8 Exponential Distribution
Figure 4-24 Lack of memory property of an Exponential
distribution.
Figure 4-24 Lack of memory property of an Exponential
distribution.
4-9 Erlang and Gamma Distributions
Erlang Distribution
The random variable X that equals the interval length
until r counts occur in a Poisson process with mean λ > 0
has and Erlang random variable with parameters λ and
r. The probability density function of X is
for x > 0 and r =1, 2, 3, ….
Erlang Distribution
The random variable X that equals the interval length
until r counts occur in a Poisson process with mean λ > 0
has and Erlang random variable with parameters λ and
r. The probability density function of X is
for x > 0 and r =1, 2, 3, ….
4-9 Erlang and Gamma Distributions
Gamma Distribution
Gamma Distribution
4-9 Erlang and Gamma Distributions
Gamma Distribution
Gamma Distribution
4-9 Erlang and Gamma Distributions
Gamma Distribution
Figure 4-25 Gamma
probability density
functions for selected
values of r and .
Gamma Distribution
Figure 4-25 Gamma
probability density
functions for selected
values of r and .
4-9 Erlang and Gamma Distributions
Gamma Distribution
Gamma Distribution
4-10 Weibull Distribution
Definition
Definition
4-10 Weibull Distribution
Figure 4-26 Weibull
probability density
functions for selected
values of and .
Figure 4-26 Weibull
probability density
functions for selected
values of and .
4-10 Weibull Distribution
4-10 Weibull Distribution
Example 4-25
Example 4-25
4-11 Lognormal Distribution
4-11 Lognormal Distribution
Figure 4-27 Lognormal probability density functions with = 0
for selected values of
2
.
Figure 4-27 Lognormal probability density functions with = 0
for selected values of
2
.
4-11 Lognormal Distribution
Example 4-26
Example 4-26
4-11 Lognormal Distribution
Example 4-26 (continued)
Example 4-26 (continued)
4-11 Lognormal Distribution
Example 4-26 (continued)
Example 4-26 (continued)
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