Continuous Probability Distributions

1
Continuous Probability Distributions

• Many continuous probability distributions,
  including
• Uniform
• Normal
• Gamma
• Exponential
• Chi-Squared
• Lognormal
• Weibull

2
Normal Distribution

• The bell-shaped curve
• Also called the Gaussian distribution
• The most widely used distribution in statistical
  analysis
• forms the basis for most of the parametric tests
  well perform later in this course.
• describes or approximates most phenomena in
  nature, industry, or research
• Random variables (X) following this distribution
  are called normal random variables.
• the parameters of the normal distribution are µ
  and s (sometimes µ and s2.)

3
Normal Distribution

• The density function of the normal random
  variable X, with mean µ and variance s2, is
• all x.

4
Standard Normal RV

• Note the probability of X taking on any value
  between x1 and x2 is given by
• To ease calculations, we define a normal random
  variable
• where Z is normally distributed with µ 0 and
  s2 1

5
Standard Normal Distribution

• Table A.1 Areas under the standard normal
  curve from - 8 to z
• Page 915 negative values for z
• Page 916 positive values for z

6
Examples

• P(Z 1)
• P(Z -1)
• P(-0.45 Z 0.36)

7
Your turn

• Use Table A.1 to determine (draw the picture!)
• 1. P(Z 0.8)
• 2. P(Z 1.96)
• 3. P(-0.25 Z 0.15)
• 4. P(Z -2.0 or Z 2.0)

8
Applications of the Normal Distribution

• A certain machine makes electrical resistors
  having a mean resistance of 40 ohms and a
  standard deviation of 2 ohms. What percentage of
  the resistors will have a resistance less than 44
  ohms?
• Solution X is normally distributed with µ 40
  and s 2 and x 44
• P(Xlt44) P(Zlt 2.0) 0.9772
• Therefore, we conclude that 97.72 will have
  a resistance less than 44 ohms. What
  percentage will have a resistance greater than 44
  ohms?

9
Terminology Used in ISE 327 Text

• A certain machine makes electrical resistors
  having a mean resistance of 40 ohms and a
  standard deviation of 2 ohms. What percentage of
  the resistors will have a resistance greater than
  44 ohms?
• Solution X is normally distributed with µ x
  40 and sx 2 and x 44
• P(Xgt44) 1 - P(Zlt 2.0) 1 - 0.9772
• Therefore, we conclude that 2.28 will have
  a resistance greater than 44 ohms.

10
Your Turn
DRAW THE PICTURE!!

• What is the probability that a single resistor
  will have a rating between 42 and 44 ohms?
• Specifications are that the resistors are 40 3
  ohms. What percentage of the resistors will be
  within specifications?

11
The Normal Distribution In Reverse

• Example
• Given a normal distribution with µ 40 and s
  6, find the value of X for which 45 of the area
  under the normal curve is to the left of X.
• If P(Z lt k) 0.45,
• k ___________
• Z _______
• X _________