Module 5.2

• Continuous random variable


  Can assume any value in an interval on the real line or in a collection of intervals
• It is not possible to talk about the probability of the random variable assuming a particular value
• Probability of the random variable assuming a value within a given interval


  The area under the graph of the probability density function between the interval
• Continuous Probability Distributions


  • The probability of the random variable assuming a value within some given interval from x1 to x2 is defined to be the area under the graph of the probability density function between x1 and x2
• Normal probability distribution


  The most important distribution for describing a continuous random variable, widely used in statistical inference
• Abraham de Moivre, a French mathematician, published The Doctrine of Chances in 1733 and derived the normal distribution
• Normal Probability Density Function


  f(x) = (1/√(2πσ²)) * e^(-(x-μ)²/2σ²)
• Normal Probability Distribution


  • Symmetric, skewness measure is zero
  • Entire family defined by mean μ and standard deviation σ
  • Highest point is at the mean, which is also the median and mode
  • Mean can be any numerical value: negative, zero, or positive
  • Standard deviation determines the width of the curve: larger values result in wider, flatter curves
• Probabilities for the normal random variable are given by areas under the curve. The total area under the curve is 1 (.5 to the left of the mean and .5 to the right)
• Empirical rule for normal distribution


  • 68.26% of values are within +/- 1 standard deviation of the mean
  • 95.44% of values are within +/- 2 standard deviations of the mean
  • 99.72% of values are within +/- 3 standard deviations of the mean
• Standard normal probability distribution


  A random variable having a normal distribution with a mean of 0 and a standard deviation of 1
• z


  The letter used to designate the standard normal random variable
• Converting to the standard normal distribution


  z = (x - μ) / σ
• NORMSINV, NORM.S.INV


  Excel functions used to compute the z value given a cumulative probability
• NORMSDIST, NORM.S.DIST


  Excel functions used to compute the cumulative probability given a z value
• Using Excel to compute standard normal probabilities


  • P(z < 1.00)
  • P(0.00 < z < 1.00)
  • P(0.00 < z < 1.25)
  • P(-1.00 < z < 1.00)
  • P(z > 1.58)
  • P(z < -0.50)
• Using Excel to find z values given probabilities

  • z value with .10 in upper tail
  • z value with .025 in upper tail
  • z value with .025 in lower tail
• Pep Zone sells auto parts and supplies including a popular multi-grade motor oil. When the stock drops to 20 gallons, a replenishment order is placed. The store manager is concerned about stockouts while waiting for replenishment.
• Demand during replenishment lead-time

  Normally distributed with mean 15 gallons and standard deviation 6 gallons
• Solving for the stockout probability

  1. Convert x to the standard normal distribution

  2. Find the area under the standard normal curve to the left of z

  3. Compute the area under the standard normal curve to the right of z
• Probability of a stockout during replenishment lead-time


  P(x > 20) = 0.2033
• Solving for the reorder point

  1. Find the z-value that cuts off an area of .05 in the right tail of the standard normal distribution

  2. Convert z.05 to the corresponding value of x
• Raising the reorder point from 20 gallons to 25 gallons decreases the probability of a stockout from about .20 to .05
• NORM.DIST


  Excel function used to compute the cumulative probability given an x value
• NORM.INV


  Excel function used to compute the x value given a cumulative probability