Statistics 400 - Lecture 6

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Statistics 400 - Lecture 6
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• Today Finished 4.5 began discrete random
  variables (5.1-5.4)
• Today Finish discrete random variables (5.5-5.7)
  and begin continuous R.V.s (6.1-6.3)
• Next Day 6.4, 6.6 and 7.1
• Assignment 2 4.14, 4.24, 4.41, 4.61, 4.79,
  5.13(a and c), 5.32, 5.68, 5.80
• Due in class Tuesday, October 2

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• Probability Model - is an assumed form of a
  distribution of a random variable

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Bernoulli Distribution

• Bernoulli distribution
• Each trial has 2 outcomes (success or failure)
• Prob. of a success is same for each trial
• Prob. of a success is denoted as p
• Prob. of a failure, q, is
• Trials are independent
• If X is a Bernoulli random variable, its
  distribution is described by
• where X0 (failure) or X1 (success)

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Example

• A backpacker has 3 emergency flares, each which
  light with probability of 0.98.
• Find probability the first flare used will light
• Find probability that first 2 flares used both
  light
• Find probability that exactly 2 flares light

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Mean and Standard Deviation

• Mean
• Standard Deviation

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Binomial Distribution

• Binomial distribution is a distribution that
  models chance variation of n repetitions of an
  experiment that has only 2 outcomes
• Random variable, X, is the count of observations
  falling in one of the categories
• X is number of successes in n Bernoulli trials
• X number of successes
• Probability of success remain constant

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• n Number of trials
• p probability of success
• X number of successes in the n trials (for X
  0,1,2,, n)
• Note

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Example 5.67

• 15 of trees in a forest have leaf damage
• If 5 are selected at random, find probability
• 3 have leaf damage
• no more than 2 have leaf damage
• 3 do not have leaf damage

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Mean and Standard Deviation

• Mean
• Standard Deviation

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Continuous Random Variables

• Can you list all points in an interval
• Have described distribution of quantitative
  variable using a histogram
• A relative frequency histogram has proportions on
  Y-axis.
• Sum of bar heights is

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• Can describe overall shape of distribution with a
  mathematical model called a density function,
  f(x)
• Describes main features of a distribution with a
  single expression
• Total area under curve is
• Area under a density curve for a given range
  gives

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• Total area under curve is

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Features

• Mode
• Median
• Quartiles
• Mean

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Standardizing

• If X is a R.V., the standardized variable, Z, has
  mean 0 and standard deviation 1.

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Normal Distributions

• Common continuous density is the normal
  distribution
• It is symmetric, bell-shaped and uni-modal
• Denoted

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• What happens if mean is changed?
• What happens if standard deviation is changed?

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Relative Location of Mean and Median

• Right Skewed
• Left Skewed
• Symmetric