statistical distributions. uniform distribution a r.v. is uniformly distributed on the interval...
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![Page 1: Statistical Distributions. Uniform Distribution A R.V. is uniformly distributed on the interval (a,b) if it probability function Fully defined by (a,b)](https://reader030.vdocuments.site/reader030/viewer/2022032604/56649e5e5503460f94b56e0e/html5/thumbnails/1.jpg)
Statistical Statistical DistributionsDistributions
Statistical Statistical DistributionsDistributions
![Page 2: Statistical Distributions. Uniform Distribution A R.V. is uniformly distributed on the interval (a,b) if it probability function Fully defined by (a,b)](https://reader030.vdocuments.site/reader030/viewer/2022032604/56649e5e5503460f94b56e0e/html5/thumbnails/2.jpg)
Uniform DistributionA R.V. is uniformly distributed on the
interval (a,b) if it probability functionFully defined by (a,b)
P(x) = 1/(b-a) for a <= x <= b = 0 otherwise
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Uniform Distribution Probability Function
1 10
1
1/9
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Probability that x is between 2 and 7.5? Probability that x = 8?
1 10
1
1/9
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Uniform DistributionThe cumulative distribution of a
uniform RV is
F(x) = 0 for x < a = (x-a)/(b-a) for a <= x
<= b = 1 otherwise
![Page 6: Statistical Distributions. Uniform Distribution A R.V. is uniformly distributed on the interval (a,b) if it probability function Fully defined by (a,b)](https://reader030.vdocuments.site/reader030/viewer/2022032604/56649e5e5503460f94b56e0e/html5/thumbnails/6.jpg)
Uniform Distribution Cumulative Function
1 10
1
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Uniform DistributionDiscrete vs. Continuous• Discrete RV
– Number showing on a die
• Continuous RV– Time of arrival – When programming, make it discrete
to some number of decimal places
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Uniform Distribution• Mean = (a+b)/2• Variance = (b-a)2 /12
• P (x < X < y) = F (y) – F (x)= (y-a) - (x-a) = y – x – a + a = y -
x b-a b-a b – a b-a
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Uniform - ExampleA bus arrives at a bus stop every 20
minutes starting at 6:40 until 8:40. A passenger does not know the schedule but randomly arrives between 7:00 and 7:30 every morning. What is the probability the passenger waits more than 5 minutes.
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Uniform Solution
5 10 15 20 25 30 40
1
1/30
X = RV, Uniform (0,30) -- i.e. 7:00 – 7:30Bus: 7:00, 7:20, 7:40Yellow Box <= 5 minute wait
A B C
P (x > 5) = A + C = 1 – B = 5/6
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Arithmetic Mean
Given a set of measurements y1, y2, y3,… yn
Mean = (y1+y2+…yn) / n
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Variance
Variance of a set of measurements y1, y2, y3,… yn is the average of the deviations of the measurements about their mean (m).
V = σ2 = (1/n) Σ (yi – m)2i=1..n
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Variance Example
Yi= 12, 10, 9, 8, 14, 7, 15, 6, 14, 10m = 10.5
V= σ2 = (1/10) ((12-10.5)2 + (10-10.5)2 +….
= (1/10) (1.52 + .52 + 1.52….) = (1/10) (88.5)
= 8.85
Standard Deviation = σ = 2.975
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Normal Distribution• Has 2 parameters
–Mean - μ–Variance – σ2
–Also, Standard deviation - σ
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Normal Dist.
0-3 -2 -1 1 2 3
Mean +- n σ
.3413
.1359
.0215
.0013
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Normal Distribution• Standard Normal Distribution has
– Mean = 0 StdDev = 1
• Convert non-standard to standard to use the tablesZ value = # of StdDev from the meanZ is value used for reading table
Z = (x – m)
σ
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Normal - ExampleThe scores on a college entrance exam
are normally distributed with a mean of 75 and a standard deviation of 10. What % of scores fall between 70 & 90?
Z(70) = (70 – 75)/10 = - 0.5Z(90) = (90 – 75)/10 = 1.5.6915 - .5 = .1915 + .9332 - .5 = .4332
= .6247 or 62.47%
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Exponential Distribution
A RV X is exponentially distributed with parameter > 0 if probability function
Mean = 1/Variance = 1 / 2 e = 2.71828182
e xP(x) =
For x >= 0
= 0 Otherwise
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Exponential Distribution
• Often used to model interarrival times when arrivals are random and those which are highly variable.
• In these instances lambda is a rate– e.g. Arrivals or services per hour
• Also models catastrophic component failure, e.g. light bulbs burning out
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Exponential Rates• Engine fails every 3000 hours
– Mean: Average lifetime is 3000 hours– = 1/3000 = 0.00033333
• Arrivals are 5 every hour– Mean: Interarrival time is 12 minutes– = 1 / 5 = 0.2
• Mean = 1 /
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Exponential Distribution
Probability Function
x
f(x)
See handout for various graphs.
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Exponential Distribution
Cumulative Function
Given Mean = 1/ Variance = 1/ 2
F(x) = P (X <=x) = 1 – e - x
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Exponential Distribution
Cumulative Function (<=)
x
1
F(x)
![Page 24: Statistical Distributions. Uniform Distribution A R.V. is uniformly distributed on the interval (a,b) if it probability function Fully defined by (a,b)](https://reader030.vdocuments.site/reader030/viewer/2022032604/56649e5e5503460f94b56e0e/html5/thumbnails/24.jpg)
Forgetfulness PropertyGiven: the occurrence of events conforms
to an exponential distribution:The probability of an event in the next x-
unit time frame is independent on the time since the last event.
That is, the behavior during the next x-units of time is independent upon the behavior during the past y-units of time.
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Forgetfulness Example
• The lifetime of an electrical component is exponentially distributed with a mean of .
• What does this mean??
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Forgetfulness Examples
The following all have the same probability
• Probability that a new component lasts the first 1000 hours.
• Probability that a component lasts the next 1000 hours given that it has been working for 2500 hours.
• Probability that a component lasts the next 1000 hours given that I have no idea how long it has been working.
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Solution to Example• Suppose the mean lifetime of
the component is 3000 hours.• = 1/3000• P(X >= 1000) = 1 – P(X <=
1000) 1 – (1-e -1/3* 1) = e -1/3 = .717
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How do we apply these?
1. We may be given the information that events occur according to a known distribution.
2. We may collect data and must determine if it conforms to a known distribution.