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Part VI: Named Continuous Random
Variables
http://www-users.york.ac.uk/~pml1/bayes/cartoons/cartoon08.jpg
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Comparison of Named Distributions
discrete
Bernoulli, Binomial,
Geometric, Negative
Binomial, Poisson,
Hypergeometric,
Discrete Uniform
continuous
Continuous Uniform,
Exponential, Gamma,
Beta, Normal
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Chapter 30: Continuous Uniform R.V.
http://www.six-sigma-material.com/Uniform-Distribution.html
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Uniform distribution: Summary
Things to look for: constant density on a line or area
Variable:
X = an exact position or arrival time
Parameter:
(a,b): the endpoints where the density is nonzero.
Density:
CDF:
0
𝑥<𝑎
1
𝑥−𝑎
𝑎
≤
𝑥
≤
𝑏
𝑎≤𝑥≤𝑏
𝑓𝑥 𝑥 = 𝑏 − 𝑎
𝐹𝑋 𝑥 =
𝑏−𝑎
0
𝑒𝑙𝑠𝑒
1
𝑏<𝑥
𝑎+𝑏
𝔼 𝑋 =
,
2
(𝑏 − 𝑎)2
𝑉𝑎𝑟 𝑋 =
12
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Example: Uniform Distribution (Class)
A bus arrives punctually at a bus stop every thirty
minutes. Each morning, a bus rider leaves her
house and casually strolls to the bus stop.
a) Why is this a Continuous Uniform distribution
situation? What are the parameters? What is X?
b) What is the density for the wait time in minutes?
c) What is the CDF for the wait time in minutes?
d) Graph the density.
e) Graph the CDF.
f) What is the expected wait time?
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Example: Uniform Distribution (Class)
A bus arrives punctually at a bus stop every thirty
minutes. Each morning, a bus rider leaves her
house and casually strolls to the bus stop.
g) What is the standard deviation for the wait
time?
h) What is the probability that the person will wait
between 20 and 40 minutes? (Do this via 3
different methods.)
i) Given that the person waits at least 15 minutes,
what is the probability that the person will wait
at least 20 minutes?
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Example: Uniform Distribution
0.04
0.03
0.02
0.01
0.00
-10
10
30
1
0.8
0.6
0.4
0.2
0
-10
10
30
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Example: Uniform Distribution (Class)
A bus arrives punctually at a bus stop every thirty
minutes. Each morning, a bus rider leaves her
house and casually strolls to the bus stop.
Let the cost of this waiting be $20 per minute plus
an additional $5.
a) What are the parameters?
b) What is the density for the cost in minutes?
c) What is the CDF for the cost in minutes?
d) What is the expected cost to the rider?
e) What is the standard deviation of the cost to the
rider?
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Chapter 31: Exponential R.V.
http://en.wikipedia.org/wiki/Exponential_distribution
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Exponential Distribution: Summary
Things to look for: waiting time until first event occurs or
time between events.
Variable:
X = time until the next event occurs, X ≥ 0
Parameter:
: the average rate
Density:
CDF:
−𝜆𝑥
𝜆𝑒
𝑓𝑥 𝑥 =
0
1
𝔼 𝑋 = ,
𝜆
𝑥>0
𝑒𝑙𝑠𝑒
−𝜆𝑥
1
−
𝑒
𝐹𝑋 𝑥 =
0
1
𝑉𝑎𝑟 𝑋 = 2
𝜆
𝑥>0
𝑒𝑙𝑠𝑒
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Example: Exponential R.V. (class)
Suppose that the arrival time (on average) of a large
earthquake in Tokyo occurs with an exponential
distribution with an average of 8.25 years.
a) What does X represent in this story? What values
can X take?
b) Why is this an example of the Exponential
distribution?
c) What is the parameter for this distribution?
d) What is the density?
e) What is the CDF?
f) What is the standard deviation for the next
earthquake?
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Example: Exponential R.V. (class, cont.)
Suppose that the arrival time (on average) of a large
earthquake in Tokyo occurs with an exponential
distribution with an average of 8.25 years.
g) What is the probability that the next earthquake
occurs after three but before eight years?
h) What is the probability that the next earthquake
occurs before 15 years?
i) What is the probability that the next earthquake
occurs after 10 years?
j) How long would you have to wait until there is a
95% chance that the next earthquake will happen?
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Example: Exponential R.V. (Class, cont.)
Suppose that the arrival time (on average) of a
large earthquake in Tokyo occurs with an
exponential distribution with an average of
8.25 years.
k) Given that there has been no large
Earthquakes in Tokyo for more than 5 years,
what is the chance that there will be a large
Earthquake in Tokyo in more than 15 years?
(Do this problem using the memoryless
property and the definition of conditional
probabilities.)
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Minimum of Two (or More)
Exponential Random Variables
Theorem 31.5
If X1, …, Xn are independent exponential random
variables with parameters 1, …, n then
Z = min(X1, …, Xn) is an exponential random
variable with parameter 1 + … + n.
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Chapter 37: Normal R.V.
http://delfe.tumblr.com/
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Normal Distribution: Summary
Things to look for: bell curve,
Variable:
X = the event
Parameters:
X = the mean
𝜎𝑋2 = 𝑡ℎ𝑒 𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒
Density:
−(𝑥−𝜇𝑥 )2 (2𝜎 2 )
𝑒
𝑓𝑥 𝑥 =
,𝑥 𝜖 ℝ
2𝜋𝜎 2
𝔼 𝑋 = X ,
𝑉𝑎𝑟 𝑋 = 𝜎𝑋2
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PDF of Normal Distribution (cont)
http://commons.wikimedia.org/wiki/File:Normal_distribution_pdf.svg
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PDF of Normal Distribution
http://www.oswego.edu/~srp/stats/z.htm
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PDF of Normal Distribution (cont)
X0
0.2
X  (2)
0.5
X0
1
X0
5
http://commons.wikimedia.org/wiki/File:Normal_distribution_pdf.svg
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Procedure for doing Normal Calculations
1) Sketch the problem.
2) Write down the probability of interest in
terms of the original problem.
3) Convert to standard normal.
4) Convert to CDFs.
5) Use the z-table to write down the values of
the CDFs.
6) Calculate the answer.
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Example: Normal r.v. (Class)
The gestation periods of women are normally
distributed with  = 266 days and  = 16 days.
Determine the probability that a gestation period is
a) less than 225 days.
b) between 265 and 295 days.
c) more than 276 days.
d) less than 300 days.
e) Among women with a longer than average
gestation, what is the probability that they give
birth longer than 300 days?
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Example: “Backwards” Normal r.v. (Class)
The gestation periods of women are normally
distributed with  = 266 days and  = 16 days. Find
the gestation length for the following situations:
a) longest 6%.
b) shortest 13%.
c) middle 50%.
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Chapter 36: Central Limit Theorem
(Normal Approximations to Discrete
Distributions – 36.4, 36.5)
http://nestor.coventry.ac.uk/~nhunt/binomial http://nestor.coventry.ac.uk/~nhunt/poisson
/normal.html
/normal.html
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Continuity Correction - 1
http://www.marin.edu/~npsomas/Normal_Binomial.htm
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Continuity Correction - 2
W~N(10, 5)
X ~ Binomial(20, 0.5)
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Continuity Correction - 3
Discrete
a<X
a≤X
X<b
X≤b
Continuous
a + 0.5 < X
a – 0.5 < X
X < b – 0.5
X < b + 0.5
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Normal Approximation to Binomial
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Example: Normal Approximation to
Binomial (Class)
The ideal size of a first-year class at a particular
college is 150 students. The college, knowing
from past experience that on the average only
30 percent of these accepted for admission
will actually attend, uses a policy of approving
the applications of 450 students.
a) Compute the probability that more than 150
students attend this college.
b) Compute the probability that fewer than 130
students attend this college.
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Chapter 32: Gamma R.V.
http://resources.esri.com/help/9.3/arcgisdesktop/com/gp_toolref
/process_simulations_sensitivity_analysis_and_error_analysis_modeling
/distributions_for_assigning_random_values.htm
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Gamma Distribution
• Generalization of the exponential function
• Uses
– probability theory
– theoretical statistics
– actuarial science
– operations research
– engineering
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Gamma Function

(t)   x e dx,t  0
t1  x
0
(t + 1) = t (t), t > 0, t real
(n + 1) = n!, n > 0, n integer
1  (2n)!

 n   

2n
2  n!2

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Gamma Distribution: Summary
Things to look for: waiting time until rth event occurs
Variable: X = time until the rth event occurs, X ≥ 0
Parameters:
r: total number of arrivals/events that you are waiting for
: the average rate
Density:
𝜆𝑟 𝑟−1 −𝜆𝑥
𝑥 𝑒
𝑓𝑥 𝑥 = Γ(𝑟)
0
𝑥>0
𝑒𝑙𝑠𝑒
𝑟−1
𝐶𝐷𝐹: 𝐹𝑋 𝑥 =
1 − 𝑒 −𝜆𝑥
𝑗=0
0
𝑟
𝑟
𝔼 𝑋 = , 𝑉𝑎𝑟 𝑋 = 2
𝜆
𝜆
(𝜆𝑥)𝑗
𝑗!
𝑥>0
𝑒𝑙𝑠𝑒
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Gamma Random Variable
k=r
𝜃=
1
𝜆
http://en.wikipedia.org/wiki/File:Gamma_distribution_pdf.svg
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Chapter 33: Beta R.V.
http://mathworld.wolfram.com/BetaDistribution.html
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Beta Distribution
• This distribution is only defined on an interval
– standard beta is on the interval [0,1]
– The formula in the book is for the standard
beta
• uses
– modeling proportions
– percentages
– probabilities
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Beta Distribution: Summary
Things to look for: percentage, proportion, probability
Variable: X = percentage, proportion, probability of interest
(standard Beta)
Parameters:
, 
Density:
𝑓𝑥 𝑥
𝛼−1
1 Γ(𝛼 + 𝛽) 𝑥 − 𝐴
=
𝐵 − 𝐴 Γ(𝛼)Γ(𝛽) 𝐵 − 𝐴
0
Density: no simple form
When A = 0, B = 1 (Standard Beta)
𝛼
𝔼 𝑋 =
, 𝑉𝑎𝑟 𝑋 =
𝛼+𝛽
𝛼+𝛽
𝑥−𝐴
𝐵−𝐴
𝛽−1
𝐴≤𝑥≤𝐵
𝑒𝑙𝑠𝑒
𝛼𝛽
2 (𝛼 + 𝛽 + 1)
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Shapes of Beta Distribution
http://upload.wikimedia.org/wikipedia/commons/9/9a/Beta_distribution_pdf.png
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X
Other Continuous Random Variables
• Weibull
– exponential is a member of family
– uses: lifetimes
• lognormal
– log of the normal distribution
– uses: products of distributions
• Cauchy
– symmetrical, flatter than normal
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Chapter 37: Summary and Review of
Named Continuous R.V.
http://www.wolfram.com/mathematica/new-in-8/parametric-probability-distributions
/univariate-continuous-distributions.html
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Summary of Continuous Distributions
Name
Density, fX(x)
Domain
CDF, FX(x)
𝔼(X)
Var(X)
Parameters
What X is
When used
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Expected values and Variances for selected
families of continuous random variables.
Family
Uniform
Exponential
Normal
Parameter(s) Expectation Variance
a,b
ab
2

1

(b  a)2
12
1
2
,2

2
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