Post on 31-Dec-2015
description
Continuous Probability Distributions Experiments can lead to continuous responses i.e.
values that do not have to be whole numbers. For example: height could be 1.54 meters etc.
In such cases the sample space is best viewed as a histogram of responses.
The Shape of the histogram of such responses tells us what continuous distribution is appropriate – there are many.
Lifetime of Component
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nsity
0.0 0.5 1.0 1.5 2.0 2.5
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Waiting TimeD
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sity
0.0 0.2 0.4 0.6 0.8 1.0
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0.4
0.6
0.8
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Normal Distribution (AKA Gaussian)• The Histogram below is symmetric & 'bell
shaped'• This is characteristic of the Normal
Distribution• We can model the shape of such a
distribution (i.e. the histogram) by a Curve
Normal Distribution The Curve may not fit the histogram
'perfectly' - but should be very close Normal Distribution - two parameters,
µ = mean, = standard deviation,
The mathematical formula that gives a bell shaped symmetric curve
f(x) = Height of curve at x =
2
2
2
)(
22
1
x
e
Normal Distribution Why Not P(x) as before?
=> because response is continuous
What is the probability that a person sampled at random is 6 foot?
Equivalent question: what proportion of people are 6 foot?
=> really mean what proportion are 'around 6 foot' ( as good as the measurement device
allows) - so not really one value, but many values close together.
Example: What proportion of graduates earn €35,000?
Would we exclude €35,000.01 or €34,999.99?
Round to the nearest €, €10, €100, €1000?
Continuous measure => more useful to get proportion from €35,000 - €40,000
Some Mathematical Jargon:
The formula for the normal distribution is formally called the normal probability density function (pdf)
The Shaded portion of the Histogram is the Proportion of interest
Can visualise this using the histogram of salaries.
Since the histogram of salaries is symmetric and bell shaped, we model this in statistics with a Normal distribution curve.
Proportion = the proportion of the area of the curve that is shaded
So proportions = proportional area under the curve = a probability of interest
Need;• To know , • To be able to find area under
curve
Area under a curve is found using integration in mathematics.
In this case would need a technique called numerical integration.
Total area under curve is 1. However, the values we need are in
Normal Probability Tables.
The Tables are for a Normal Distribution with = 0 and = 1
• this is called the Standard Normal• Can 'convert' a value from any normal to the
standard normal using standard scores (Z scores)
Value from any NormalDistribution
Standardise
Corresponding Value from
Normal = 0 = 1
Z :score edstandardis
x
Standard Normal
Z scores are a unit-less quantity, measuring how far above/below a certain score (x) is, in standard deviation units.
Example: A score of 35, from a normal distribution with
= 25 and = 5.Z = ( 35 − 25) / 5 => 10/5 = 2
So 35 is 2 standard deviation units above the mean
What about a score of 20 ?
Z = ( 20 - 25) / 5 => − 5 / 5 = − 1
So 20 is 1 standard unit below the mean
Z-Score Example
Positive Z score => score is above the meanNegative Z score => score is below the mean
By subtracting and dividing by the we convert any normal to = 0, =1, so only need one set of tables!
Z-Score Example
From looking at the histogram of peoples weekly receipts, a supermarket knows that the amount people spend on shopping per week is normally distributed with:
= €58 = €15.
Example:
What is the probability that a customer sampled at random will spend less than €83.50 ?
Z = ( x − ) / = ( €83.50 - €58 ) / €15 => 1.7
Area from Z=1.7 to the left can be read in tables
From tables area less than Z = 1.7 => 0.9554
So probability is 0.9554 Or 95.54%
What is the probability that a customer sampled at random will spend more than €83.50 ?
Z = ( x − ) / = ( €83.50 - €58 ) / €15 => 1.7
Area from Z=1.7 to the right can be read in tables
From tables area greater than Z = 1.7 => 1- 0.9554 = 0.0446
So probability is 0.0446 Or 4.46%
Exercise Find the proportion of people who spend
more than €76.75 Find the proportion of people who spend
less than €63.50
Note: The tables can also be used to find other areas
(less than a particular value, or the area between two points)
Characteristics of Normal Distributions Standard Deviation has particular
relevance to Normal distribution Normal Distribution => Empirical Rule
68%
95%
99.7% Between Z (lower, upper)
%Area
-1,1 68 %
-2,2 95 %
-3,3 99.7 %
-∞, +∞ 100%
The normal distribution is just one of the known continuous probability distributions.
Each have their own probability density function, giving different shaped curves.
In each case, we find probabilities by calculating areas under these curves using integration.
However, the Normal is the most important – as it plays a major role in Sampling Theory.
Other important continuous probability distributions include• Exponential distribution – especially positively
skewed lifetime data.
• Uniform distribution.
• Weibull – especially for ‘time to event’ analysis.
• Gamma distribution – waiting times between Poisson events in time etc.
• Many others…..
Summary – Random Variables There are two types – discrete RVs and
continuous RVs
For both cases we can calculate a mean (μ) and standard deviation (σ)
μ can be interpreted as average value of the RV
σ can be interpreted as the standard deviation of the RV
Summary Continued… For Discrete RV we often have a mathematical formula
which is used to calculate probabilities,
i.e. P(x) = some formula
This formula is called the Probability Mass Function (PMF)
Given the PMF you can calculate the mean and variance by:
When the summation is over all possible values of x
222 )(
)(
xPx
xxP
Summary Continued… For continuous RVs, we use a Probability Density
Function (PDF) to define a curve over the histogram of the values of the random variables.
We integrate this PDF to find areas which are equal to probabilities of interest.
Given the PDF you can calculate the mean and variance by:
Where f(x) is usual mathematical notation for the PDF
-dx )(dx )( 22 xfxxxf