ENGR 2600
Modeling and Analysis of
Uncertainty
Handout 3
Probability Distributions for
Continuous Random Variables
1
A Lottery
What is expected return from this?
• $20: 5%
• $100: 0.5%
• $1000: 0.01%
 $1
 $.5
 $.1
Expected return (mean) vs. average
At what price will you sell it?
How can you make your lottery business more popular?
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2
Powerball change
In the old version (59 White balls + 35 PB), the chance of jackpot
= 1 / (59C5 x 35C1) = 1 / 175,223,510
In the new version (69 WB + 26 PB),
 1 / (69C5 x 26C1) = 1 / 292,201,338
http://www.durangobill.com/PowerballOdds.html
Don’t forget they know PROBABILITY!
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Probability Distributions for Continuous Variables
Probability histogram of
depth measured to the
nearest meter
Depth measured to the
nearest centimeter
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Probability density function
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Constraints
Continuous
Discrete
•Non-negative:
•Integrates to 1:
Example 3.1
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Example 3.2
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Cumulative distribution function
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Example 3.3
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Example 3.4
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Example 3.5
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Martingale bet
•
•
•
•
“Doubling of your betting when you loose”
The strategy had the gambler double his bet after every loss so that
the first win would recover all previous losses plus win a profit
equal to the original bet.
Requires infinite time and money for making it sure thing
Think why Casino imposes betting limits
Popular in
18th-century France
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Mean and Variance
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Expectations
Continuous
Discrete
•Expectation of a function g(X):
•Variance and SD:
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Example 3.6
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Example 3.7
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Continuous uniform distribution
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Expectation and Variance of
Uniform Distribution
•E(X) = ∫ab x/(b-a) dx = ½ (b+a)
a
b
½ (b+a)
•E(X2) = ∫ab x2/(b-a) dx = 1/3 (b2+ba+a2)
Var(X) = E(X2) – E(X)2 = 1/3 (b2+ba+a2) – ¼(b2+2ba+a2)
= 1/12 (b-a)2
Example 3.9
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Normal distribution
Found by Carl Friedrich Gauss
The normal distribution is a two-parameter distribution
for continuous RV’s with a symmetric, bell-shaped
distribution
f ( x)
1
e
σ 2π
1  x−µ 
− 

2 σ 
2
-∞ ≤ x ≤ ∞
E ( x) = µ
V ( x) = σ 2
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Normal distribution
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Derivation?
https://www.youtube.com/watch?v=ebewBjZmZTw
You don’t have to memorize the formula here
Standard Normal Distribution has way easier form
Mathematically, in normally distributed data, it is observed that the rate at which the
frequencies fall off is proportional to :
• distance of the particular value from the mean,
• the frequencies themselves
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Standardization
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Z
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Standardization of the
Normal Random Variable
Z=
X −µ
σ
= Expresses object in terms of the number of
standard deviations from mean.
X ~ N ( µ , σ ) ⇒ Z ~ N (0, σ = 1)
f(Z)
CDF
Standard Normal Distribution
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Formatting of the Normal Table
XZ, get Φ(z)
z = (x-μ)/σ
μ: Expected/mean value
σ: Standard deviation
Left column: accuracy of z to one decimal place
First row: accuracy of z to two decimal places
Table in LMS
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Example 3.11
Suppose X ∼ N {µ, σ2} and Z ∼ N {0, 1}
Determine the following probabilities:
(a) P (Z > 3);
(b) P (0 < Z < 1);
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(c) P (−1 < Z < 1);
(d) P (−1 < Z < 1.5).
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Example 3.12
Determine the value of z that solves each of the following:
(a) P (−z < Z < z) = 0.95
(b) P (−1 < Z < z) = 0.5.
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Example 3.13
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Example 3.14
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Example 3.15
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Approximating Binomial by Normal
• As np and nq increase, B(n,p)  N(np,npq)
Don’t forget μ=np and σ2=npq=np(1-p)
• Rule of thumb: np>10 and nq>10
n = 6 and p = 0.5
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Exponential Distribution
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f ( x ) = λe
− λx
x≥0
x
F ( x) = ∫ f ( x)dx
∫ λe
0
𝑥𝑥
− λx
= −e
− λx
𝑥𝑥
−𝜆𝜆𝜆𝜆
−𝜆𝜆𝜆𝜆
= � 𝜆𝜆𝑒𝑒
𝑑𝑑𝑑𝑑 = −𝑒𝑒
�
0
= −e
− λx
[
− −e
F ( x) = 1 − e
−λ 0
]
0
− λx
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E ( x) = 1
λ
V ( x) = 1 2
λ
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Example 3.19
User log-ons to a large computer network system follow a Poisson process that has a
mean of 25 log-ons per hour.
(a)
Determine the probability that there are no log-ons in a 6 minute interval.
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Poisson vs Exponential
Suppose # of events occurring in unit time interval has a
Poisson distribution with parameter λ0 , and t1 is the time
until the first event occurs.
Probability that the event occurs before specific time t:
P(t1 ≤ t) = 1 – P [no events in (0, t)]
p ( x) = e − λ λx / x! x = 0,1,2......∞
= 1 – P [ x=0; λ=λ0 t ]
𝑒𝑒 −𝜆𝜆0𝑡𝑡 (𝜆𝜆0𝑡𝑡)0
=1−
= 1 − 𝑒𝑒 −𝜆𝜆0𝑡𝑡
0!
Poisson
The cdf of the exponential distribution 𝐹𝐹(𝑡𝑡) = 1 − 𝑒𝑒 −𝜆𝜆𝑡𝑡
(the probability that event happens between 0 to t)
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User log-ons to a large computer network system follow a Poisson process that has a
mean of 25 log-ons per hour.
(a)
.
(b)
Determine the probability that the next log-on is within 2 to 3 minutes.
(c)
What is the interval of time such that no log-ons occur is 90%?
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Example 3.21
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Maximum Entropy Distribution
• Entropy
= a measure of disorder, randomness, or uncertainty
• Among all continuous probability distributions with mean
μ, the exponential distribution with λ = 1/μ has the largest
differential entropy
• Many events in nature follows Poisson process (random)
• If question is about number  Poisson distribution
• If question is about ‘distance between events’  Exponential
distribution
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Memoryless
We leave for a service period of s hours and then return to
find the component still working; what now is the probability
that it lasts at least an additional t hours?
In symbols, we wish to know P(X ≥ s+t | X ≥ s)
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Summary
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