Download Normal Distribution

Continuous Probability Distributions
Continuous Probability Distributions
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Density
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Density
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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.
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Lifetime of Component
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Waiting Time
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Normal Distribution (AKA Gaussian)
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•
•
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
( x   )2
1
f(x) = Height of curve at x =
2 
2
e
2 2
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)
Can visualise this using the histogram of salaries.
The Shaded
portion of the
Histogram is the
Proportion of
interest
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.
Standard Normal
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
Normal
Distribution
Standardise
standardis ed score : Z 
x

Corresponding
Value from Normal
=0
=1
Z-Score Example
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 mean
Negative 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!
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.
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
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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
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
Standard Deviation has particular relevance to
Normal distribution
Normal Distribution => Empirical Rule
Between Z
%Area
99.7%
(lower, upper)
-1,1
68 %
-2,2
95 %
-3,3
99.7 %
-∞, +∞
100%
68%
95%

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:
   xP( x )
   x P( x )  
2

2
2
When the summation is over all possible values of x
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:

   xf ( x) dx



   x f ( x)dx - 
2
2

Where f(x) is usual mathematical notation for the PDF