730 words (approximately 4 mins)

Normal Distribution

Notes:

  • Used for sums of random variables
  • Characterized by 2 numbers:
    • mean (mu)
    • standard deviation (sigma-squared)
  • They have continuous outcomes (weight, cost, time, etc)

Normal PDFs with different means and variances

Example:

  • 95% prediction interval for normal distribution:
    • Problem: Each customer spends a random amount of money, mean = $25, SD = $5. What is a 95% PI for the amount spent by a randomly selected customer?
    • Solution: mean +/- 2 * sd = 25 +/- 2 * 5, so 95% PI is [$15, $35]
  • 95% of normal probability density function is within 1.96 * SD of its mean.
    • 1.96 is often rounded to 2.

Notes:

  • Prediction Interval (PI) – an estimate of an interval in which a future observation will fall, with certain probability, given what has already been observed.
  • Confidence Interval – an estimated interval within which an unknown parameter may plausibly lie.
    • It describes ranges of parameter values that are consistent with data.
    • Confidence Level – a probability with which an estimated interval will plausibly contain the value of a parameter.
dnorm(x, mean=n, sd=m)	# returns the probability density function (PDF) at x for a normal 
				        # distribution with parameter values specified for its mean and 
				        # standard deviation.
pnorm(x, mean=n, sd=m) 	# returns the cumulative distribution function (CDF); used to find 
				        # probabilities that the outcome falls in a specified interval.

Inverse-CDF Function (Quantile Function) (Q) (qnorm)

Notes:

  • Inverse functions start on the Y axis and read off of the X axis to find the value for the input on the Y axis.
qnorm(p, mean, sd) # returns value Q such that fraction p of a normal distribution is less than Q.
  • Q becomes the p-th “quantile”
  • This is the inverse of the CDF (pnorm())

Example:

  • A customer’s spend is normally distributed with mean = $25, sd = $5. What dollar amount do 99% of customers spend more than?
qnorm(0.5, 0, 23)

13.37

Example (95% prediction interval for normal distribution)

  • Quesetion: Each customer at a store spends a random amount of money with mean = $25 and SD = $5. What is a 95% prediction interval (PI) for the amount spent by a randomly selected customer?

mean +/- 2 * SD = 25 +/- 2 * 5, so 95% PI is [$15, $35] (this uses 2 as an approximation for 1.96)

Example (pnorm)

  • Question: If a customer’s expenditures are normally distributed with mean = $25 and SD = $5, what is the probability that the customer spends less than $20?
pnorm(20, 25, 5)

0.1586553

Practice

Practice with pnorm()

  • Q: The waiting time at a shop is approximately normally distributed with mean = 75 minutes and SD = 25 mins. What is the probability that a customer’s waiting time will exceed 90 minutes?
  • A: 1 - pnorm(90, 75, 25) = 0.226
    • Q: What fraction of customers are predicted to have waiting times of less than 60 minutes?
    • A: pnorm(60, 75, 25) = 0.226 § Q: What is the waiting time that only 5% of customers exceed? § A: qnorm(0.95, 75, 20) = 107.9 minutes
  • Q: An insurance pool pays a random number of claims per year. Suppose that the number of claims is modeled as a normal distribution with mean = 600 and SD = 15.49. What is the probability that the number of claims is less than 630?
  • A: pnorm(630, 600, 15.49) = 0.973
  • Q: A gas station seels a random amount of gas per day. Amount sold per day has a normal distribution with mean = 5400 gallons and SD = 150 gallons. How much gasoline should the station “stock” at the start of the day to keep the probability of running out by the end of the day <= 2%?
  • A: qnorm(0.98, 5400, 150) = 5708.06 gallons. If the gas station stocks 5708.06 gallons, there is a 98% probability that daily demand will be less than this, and hence a 2% probability of running out.

Practice CDF calculations

  • Q: Finish times for runners a normally distributed with mean = 200 mins, SD = 30 mins. What is the probability that a runner will complete the marathon within 3 hours?
  • A: pnorm(180, 200, 30) = 0.252
    • Q: By what time will 10% of runners have completed the marathon?
    • A: qnorm(0.10, 200, 30) = 161.55 § Q: What fraction of runners will complete the marathon between 3 and 4 hours? § A: pnorm(240, 200, 30) - pnorm(180, 200, 30) = 0.656