Random Variables and Probability Distributions

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Random Variables and Probability Distributions

• Random Variables - Random responses corresponding
  to subjects randomly selected from a population.
• Probability Distributions - A listing of the
  possible outcomes and their probabilities
  (discrete r.v.s) or their densities (continuous
  r.v.s)
• Normal Distribution - Bell-shaped continuous
  distribution widely used in statistical inference
• Sampling Distributions - Distributions
  corresponding to sample statistics (such as mean
  and proportion) computed from random samples

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Normal Distribution

• Bell-shaped, symmetric family of distributions
• Classified by 2 parameters Mean (m) and standard
  deviation (s). These represent location and
  spread
• Random variables that are approximately normal
  have the following properties wrt individual
  measurements
• Approximately half (50) fall above (and below)
  mean
• Approximately 68 fall within 1 standard
  deviation of mean
• Approximately 95 fall within 2 standard
  deviations of mean
• Virtually all fall within 3 standard deviations
  of mean
• Notation when Y is normally distributed with mean
  m and standard deviation s

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Normal Distribution
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Example - Heights of U.S. Adults

• Female and Male adult heights are well
  approximated by normal distributions
  YFN(63.7,2.5) YMN(69.1,2.6)

Source Statistical Abstract of the U.S. (1992)
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Standard Normal (Z) Distribution

• Problem Unlimited number of possible normal
  distributions (-? lt m lt ? , s gt 0)
• Solution Standardize the random variable to have
  mean 0 and standard deviation 1

• Probabilities of certain ranges of values and
  specific percentiles of interest can be obtained
  through the standard normal (Z) distribution

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Standard Normal (Z) Distribution

• Standard Normal Distribution Characteristics
• P(Z ? 0) P(Y ? m ) 0.5000
• P(-1 ? Z ? 1) P(m-s ? Y ? ms ) 0.6826
• P(-2 ? Z ? 2) P(m-2s ? Y ? m2s ) 0.9544
• P(Z ? za) P(Z ? -za) a (using Z-table)

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Finding Probabilities of Specific Ranges

• Step 1 - Identify the normal distribution of
  interest (e.g. its mean (m) and standard
  deviation (s) )
• Step 2 - Identify the range of values that you
  wish to determine the probability of observing
  (YL , YU), where often the upper or lower bounds
  are ? or -?
• Step 3 - Transform YL and YU into Z-values

• Step 4 - Obtain P(ZL? Z ? ZU) from Z-table

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Example - Adult Female Heights

• What is the probability a randomly selected
  female is 510 or taller (70 inches)?
• Step 1 - Y N(63.7 , 2.5)
• Step 2 - YL 70.0 YU ?
• Step 3 -

• Step 4 - P(Y ? 70) P(Z ? 2.52) .0059 ( ?
  1/170)

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Finding Percentiles of a Distribution

• Step 1 - Identify the normal distribution of
  interest (e.g. its mean (m) and standard
  deviation (s) )
• Step 2 - Determine the percentile of interest
  100p (e.g. the 90th percentile is the cut-off
  where only 90 of scores are below and 10 are
  above)
• Step 3 - Turn the percentile of interest into a
  tail probability a and corresponding z-value
  (zp)
• If 100p ? 50 then a 1-p and zp za
• If 100p lt 50 then a p and zp -za
• Step 4 - Transform zp back to original units

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Example - Adult Male Heights

• Above what height do the tallest 5 of males lie
  above?
• Step 1 - Y N(69.1 , 2.6)
• Step 2 - Want to determine 95th percentile (p
  .95)
• Step 3 - Since 100p gt 50, a 1-p 0.05
• zp za z.05 1.645
• Step 4 - Y.95 69.1 (1.645)(2.6) 73.4

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Statistical Models

• When making statistical inference it is useful to
  write random variables in terms of model
  parameters and random errors

• Here m is a fixed constant and e is a random
  variable
• In practice m will be unknown, and we will use
  sample data to estimate or make statements
  regarding its value

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Sampling Distributions and the Central Limit
Theorem

• Sample statistics based on random samples are
  also random variables and have sampling
  distributions that are probability distributions
  for the statistic (outcomes that would vary
  across samples)
• When samples are large and measurements
  independent then many estimators have normal
  sampling distributions (CLT)
• Sample Mean
• Sample Proportion

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Example - Adult Female Heights

• Random samples of n 100 females to be selected
• For each sample, the sample mean is computed
• Sampling distribution

• Note that approximately 95 of all possible
  random samples of 100 females will have sample
  means between 63.0 and 64.0 inches