Ch5. Probability Densities II

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Ch5. Probability Densities II

• Dr. Deshi Ye
• yedeshi_at_zju.edu.cn

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5.4 Other Prob. Distribution

• Uniform distribution equally likely outcome

Mean of uniform
Variance of uniform
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Ex.

• Students believe that they will get the final
  scores between 80 and 100. Suppose that the final
  scores given by the instructors has a uniform
  distribution.
• What is the probability that one student get the
  final score no less than 85?

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Solution
f(x)
0.05
x
80
100
85

• P(85 ? x ? 100) (Base)(Height)
• (100 - 85)(0.05) 0.75

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5.6 The Log-Normal Distr.

• Log-Normal distribution

It has a long right-hand tail
By letting ylnx
Hence
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Mean of Log-Normal

• Mean and variance are

Proof.
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Gamma distribution
Mean and Variance
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The Exponential Distribution

• By letting in the Gamma distribution

Mean and Variance
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5.8 The Beta Distribution

• When a random variables takes on values on the
  interval 0,1

Mean and Variance
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5.9 Weibull Distribution
Mean and Variance
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5.10 Joint distribution

• Experiments are conduced where two or more random
  variables are observed simultaneously in order to
  determine not only their individual behavior but
  also the degree of relationship between them.

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Two discrete random variables
The probability that X1 takes value x1 and X2
will take the value x2
EX.
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Marginal probability distributions
EX.
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Conditional Probability distribution
The conditional probability of X1 given that
X2x2
If two random variables are independent
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EX.

• With reference to the previous example, find the
  conditional probability distribution of X1, given
  that X21. Are X1 and X2 independent?
• Solution.

Hence, it is dependent
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Continuous variables

• If are k continuous random
  variables, we refer to as the
  joint probability density of these random
  variables

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EX.

• P179.

Find the probability that the first random
variable between 1 and 2 and the second random
variable between 2 and 3
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Marginal density

• Marginal density of X1

Example of previous
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Distribution function
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Independent
If two random variables are independent iff the
following equation satisfies.
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Properties of Expectation

• Consider a function g(x) of a single random
  variable X. For example g(x) 9x/5 32.
• If X has probability density f(x), then the mean
  or expectation of g(x) is given by

Or
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Properties of Expectation
If a and b are constants
Proof. Both in continuous and discrete case
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Covariance

• Covariance of X1 and X2 to measure

Theorem. When X1 and X2 are independent, their
covariance is 0
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5.11 Checking Normal

• Question A data set appears to be generated by a
  normal distributed random variable
• Collect data from students last 4 numbers of
  mobiles

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Simple approach

• Histogram can be checked for lack of symmetry
• A single long tail certainly contradict the
  assumption of a normal distribution

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Normal scores plot

• Also called Q-Q plot, normal quantile plot,
• normal order plot, or rankit plot.
• Normal scores an idealized sample from the
  standard normal distribution. It consists of the
  values of z that divide the axes into equal
  probability intervals. For example, n4.

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Steps to construct normal score plot

• 1) order the data from smallest to largest
• 2) Obtain the normal scores
• 3) Plot the i-th largest observation, versus i-th
  normal score mi, for all i.
• Plot

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Normal scores in Minitab

• In minitab, the normal scores are calculated in
  different ways
• The i-the normal score is

Where is the inverse cumulative
distribution function of the standard normal
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Property of Q-Q plot

• If the data set is assumed to be normal
  distribution, then normal score plot will
  resemble to a /line through the original.

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5.12 Transform observation to near normality

• When the histogram or normal scores plot indicate
  that the assumption of a normal distribution is
  invalid, transformations of the data can often
  improve the agreement with normality.

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Simulation

• Suppose we need to simulate values from the
  normal distribution with a specified

The value x can be calculated from the value of a
standard normal variable z

• From

• z can be obtained from the value for a uniform
  variable u by numerically solving uF(z)
• Box-Muller-Marsaglia method it starts with a
  pair of independent variable u1 and u2, and
  produces two standard normal variables

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Box-Muller-Marsaglia
It starts with a pair of independent variable u1
and u2, and produces two standard normal
variables
Then
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Simulation from exponential distribution

• Suppose we wish to simulate an observation from
  the exponential distribution

The computer would first produce the value u from
the uniform distribution. Then
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• Population and sample

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Population and Sample

• Investigating a physical phenomenon, production
  process, or manufactured unit, share some common
  characteristics.
• Relevant data must be collected.
• Unit the source of each measurement.
• A single entity, usually an object or person
• Population entire collection of units.

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Population and sample
Population
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Key terms

• Population
• All items of interest
• Sample
• Portion of population
• Parameter
• Summary Measure about Population
• Statistic
• Summary Measure about sample

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Examples
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Sample

• Statistical population the set of all
  measurement corresponding to each unit in the
  entire population of units about which
  information is sought.
• Sample A sample from a statistical population is
  the subset of measurements that are actually
  collected in the course of investigation.

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Sample

• Need to be representative of the population
• To be large enough to contain sufficient
  information to answer the question about the
  population

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Discussion

• P10, Review Exercises 1.2
• A radio-show host announced that she wanted to
  know which singer was the favorite among college
  students in your school. Listeners were asked to
  call and name their favorite singer. Identify the
  population, in terms of preferences, and the
  sample.
• Is the sample likely to be more representative?
• Comment. Also describe how to obtain a sample
  that is likely to be more representative.