Todays Goals

1
Todays Goals

• Review
• Final Tuesday May 19, 800 am. ELab 323.
• page of notes, front and back. I will supply
  Normal and t-tables
• Mini-Project due May 15.
• Work with a partner. Each pair turn in one
  co-written report.
• Sample final and solutions are posted.
• Recommended practice problems 6.9 7.13, 7.33c,
  7.35a, 7.37a 8.29a, 8.31, 8.35, 8.53 9.3

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Topics on Final

• Calculating Probabilities
• Using independence
• Mutually exclusive
• Conditional probability
• Flaw of Averages
• Applying Bayes Rule and Total Probability

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Topics on Final

• Applying Probability Models
• Discrete Models
• Binomial
• HyperGeometric
• Negative Binomial
• Poisson
• Continuous
• Uniform
• Normal
• Exponential

4
Topics on Final

• Joint Probability
• Covariance
• Correlation
• Joint probability Distributions
• Means of a function of variables
• The distribution of sample means
• Confidence Intervals
• Hypothesis Testing

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Joint Probability Density Function
Let X and Y be continuous rvs. Then f (x, y) is
a joint probability density function for X and Y
if for any two-dimensional set A
If A is the two-dimensional rectangle
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Marginal Probability Density Functions
The marginal probability density functions of X
and Y, denoted fX(x) and fY(y), are given by
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Independence

• Two discrete R.V. are independent if
• p(x,y) p(x)p(y)
• Two continuous random variables X and Y are said
  to be independent if for every pair of x and y
  values,
• f(x,y) fX(x) fY(y).

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Example

• f(x,y) x y for 0x,y1
• Is this a joint pdf?
• yes
• What is the marginal pdf of x?

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Example

• f(x,y) x y for 0x,y1
• Is this a joint pdf?
• yes
• What is the marginal pdf of x?
• True or False X and Y are independent.

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Joint Probability -- Covariance

• Covariance is a measure of how related two
  variables are.
• Cov(X,Y) EX-mxEY-my
• If X and Y are independent

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Joint probability -- Correlation

• The correlation Corr(X,Y) between two random
  variables X and Y is
• This number is always between -1 and 1
• If X and Y are independent, Corr(X,Y) 0
• However, the converse is not true

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Expected Value
If X and Y are independent random variables,
then EXY EXEY.
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Find Ex-y
EX 5.55 EY 7.4
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Find Ex-y
EX 5.55 EY 7.4
Ex-y .02 0 .065 .1210 .045.150
.35.0210.155.140 3.4 (not equal to 7.4
5.55 1.85)
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Sample Means

• Before you take a sample, you have a probability
  distribution over the sample mean.
• After you take a sample, you just have a number.
• We use the a priori probability distribution in
  order to figure out something about the
  population based on the sample mean.

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Sample Mean
Let X1,, Xn be a random sample from a
distribution with mean value and standard
deviation Then is
the sample mean.
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Sample Mean
Let X1,, Xn be a random sample from a
distribution with mean value and standard
deviation Then is
the sample mean.
This is always true regardless of the population
density.
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Sample Mean - Example

• We take 3 samples from an exponential
  distribution with parameter l2.
• What is

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Sample Mean - Example

• We take 3 samples from an exponential
  distribution with parameter l2.
• What is

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Sample Mean
Let X1,, Xn be a random sample from a
distribution with mean value and standard
deviation Then is
the sample mean.
What does the CLT tell us about the distribution
of the sample mean when the sample is large?
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Sample Mean
Let X1,, Xn be a random sample from a
distribution with mean value and standard
deviation Then is
the sample mean. When n is large (over 30 or so)
then the sample mean is approximately normal with
mean m and standard deviation
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Sample Mean - Example

• We take 50 samples from an exponential
  distribution with parameter l2.
• What is p(Xbar gt.55)?

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Sample Mean - Example

• We take 50 samples from an exponential
  distribution with parameter l2.
• What is p(Xbar gt.55)?

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Sample Mean - Example

• We take 50 samples from an exponential
  distribution with parameter l2.
• What is p(Xbar gt.55)?

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Confidence Interval

• We often want to estimate the actual population
  mean based on the sample mean.
• It is unlikely that the sample mean will be
  exactly equal to the population mean.
• Confidence intervals give us some idea of how
  likely it is that the population is near the
  sample mean.

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Confidence Interval
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Hypothesis Testing sample mean
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Hypothesis Testing

• H0 mean m0
• Our test statistic is

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Notes

• total probability formula
• Bayes rule formula
• Probability models formulas, means and variances
• covariance and correlation formulas
• distribution of sample mean
• Formulas for confidence intervals
• formulas for hypothesis testing