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STAT131 Week 6 L1b Introduction to Continuous Random Variables. Anne Porter alp@uow.edu.au. Sample. Population Model Discrete Random Variable. Review: Sample and Model Centre and spread . Centre Spread Variance Spread Standard Deviation. S x 2. S x. Binomial~ (n,p).

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stat131 week 6 l1b introduction to continuous random variables

STAT131 Week 6 L1b Introduction to Continuous Random Variables

Anne Porter

alp@uow.edu.au

review sample and model centre and spread
Sample

Population Model

Discrete Random Variable

Review: Sample and ModelCentre and spread

Centre

Spread

Variance

Spread

Standard

Deviation

Sx2

Sx

review discrete random variables specific case of the binomial
Binomial~ (n,p)

Population Model

DR Variable

Review Discrete Random Variables & Specific case of the Binomial

Centre

Spread

Variance

Model Fit

Calibration

Poisson( )

E(X)=np

Var(X)=np(1-p)

discrete versus continuous
Discrete Versus Continuous
  • Discrete measurement
  • Continuous

There is no possible measure between the points on the scale

Examples of discrete random variables:

The number of successes in n fixed trials - Binomial

The count of some event per unit time

The set of possible values of the RV is neither finite

not countably infinite.

examples continuous rv
Examples: Continuous RV
  • What is a sample space?

Sample space is the values that the random variable can

possibly take on.

  • What is the sample space that the life of a battery can take on?

That is the battery does not ever work,

0 units of time, to possibly working for ever

features of models
Features of models
  • What features of models are we interested in?
  • Shape, probability of some event happening
  • Centre
  • Spread
  • Fit of data to a model
  • Calibration of the model

Note that in some instances we don’t actually

need to estimate spread eg Poisson where Mean = Variance

probability for continuous random variables
Probability for Continuous Random Variables
  • With our infinitely divisible measure of life time of the battery we are not interested in the probability of a precise individual outcome.
  • Example: P(X=5.3456743234456...units of time)
  • P(X=a)=0 for any constant a. The probability of any precise value is approximately zero.
  • The data are continuous, infinitely divisible

Hence we are interested in intervals of values

Example: P( 10 <X < 20)

finding probability
Finding probability

Is there any difference in answer for these

  • P( 10 <X < 20)
  • P( 10 <X < 20)
  • P( 10 <X < 20)
  • P( 10 < X < 20)

No because the probability of exactly equalling a constant

given a continuous random variable is

zero

finding probability continuous random variables
Finding probability: Continuous Random Variables
  • How can we find a estimate of the probability function of the continuous random variable, the lifetime of batteries?

Test the lifetime of a number of batteries , that is, use the relative frequency approach.

finding probability continuous random variables10
Finding probability: Continuous Random Variables
  • How could we visually represent the relative frequency of the random variable?

A histogram

what happens to the histogram as n increased

As n increases

What happens to the histogram as n increased
  • The Rel Freq(a<X<b)-----------> Prob(a<X<b)

The curve becomes a smooth line

Different random variables have different probabilities

associated with their events and hence the different probability density functions have different shapes.

finding probability continuous random variables12
Finding probability: Continuous Random Variables
  • How do we find the area in the histogram?

By summing the areas of all the individual bars

  • How do we find the area under a curve?

By finding the area of the smallest possible bars and as the

size decreases it becomes finding the integral over the

domain

what were the conditions for defining a discrete random variable
What were the conditions for defining a discrete random variable?
  • All px>0 and all px<1
  • Sum of all px=1
conditions defining a continuous random variable
Conditions defining a Continuous Random Variable
  • From the 'area =probability' property the following hold
      • f(x)> 0 for all and,
      • the total area is 1, ie.
      • the probability of a value between some interval a

and b is given by

cumulative probability function
Cumulative probability function
  • Just as for a discrete random variable we could accumulate the probabilities to produce a cumulative probability function we can do the same for continuous random variables. It is the probability that X has a value less than or equal to x. This is expressed as F(x)=P(X<x) where thethe area to the left of x is given by
  • At x= the area is 0 and at x= the area is 1.
a comparison between discrete and continuous random variables
A comparison between discrete and continuous random variables.
  • Discrete random variables takes on a finite number or countably infinite number of values, whilst
  • Continuous random variablesconsists of one or more (possibly infinite) intervals on the real line
a comparison between discrete and continuous random variables18
A comparison between discrete and continuous random variables.
  • For discrete random variables the probability of outcome x is given by P(X=x) or px
  • For a continuous r.v P(X=a)=0 for any constant a. Hence we are interested in intervals of values P(a<X<b), eg P( 10 <X < 20).
  • For continuous random variables probability is determined for an interval eg for outcome a to b
an example
An example
  • For the random variable with

the pdf defined as illustrated, y=1/2 0<x<2. Find the area under the function by using

    • Area=Length by Breadth
    • Integration

Area=2x0.5=1 square unit

an example20

1

An example
  • What is the probability of P(0<X<1)?

0.5

How did we do this?

By use of symmetry

centre
Centre
  • The sample mean can provide an estimate of the population mean, E(X).
  • In both discrete and continuous cases this is provided by
population mean for random variables
Population mean for Random Variables
  • The population mean symbolised

is found for

      • discrete random variables by

and for

      •  continuous random variables
an example23
An example
  • What mean for this distribution

1

How did we do this?

By use of symmetry

How do we calculate the mean

For a discrete random variable?

How then do we calculate the mean

for a continuous random variable?

spread
Spread
  • The sample variance is s2This can be used as an estimate of
  • For both discrete and continuous r.v this is
spread25
Spread
  • For discrete random variables this is given by

or

  • For continuous random variables this is given by

or

goodness of fit

Continuous Random variable

Histogram (observed & expected )

- The bins can have different widths

- The bins can start in different positions

- This will affect the numbers

and the numbers expected in cells

- The expected count in each > 5

  • Discrete Random Variable
  • Bar Chart
    • observed &
    • expected counts
Goodness of fit
goodness of fit27
Goodness of Fit
  • To see if the data observed fit a given model
  • Calculate
  • For discrete models d=
  • For continuous models

d=(g-p-1)

d=g-p-1 where p is the number of parameters estimated

So for the Poisson () we have one parameter 

we will continue
We will continue ...
  • Looking at two specific continuous distributions